PrefacetotheFirstRussianEdition
Theprefacetothefirstpartcontainedaratherdetailedcharacterizationofthecourse asawhole,andhenceIconfinemyselfheretosomeremarksonthecontentofthe secondpartonly.
Thebasicmaterialofthepresentvolumeconsistsontheonehandofmultipleintegralsandlineandsurfaceintegrals,leadingtothegeneralizedStokes’formulaand someexamplesofitsapplication,andontheotherhandthemachineryofseriesand integralsdependingonaparameter,includingFourierseries,theFouriertransform, andthepresentationofasymptoticexpansions.
Thus,thisPart2basicallyconformstothecurriculumofthesecondyearofstudy inthemathematicsdepartmentsofuniversities.
Soasnottoimposerigidrestrictionsontheorderofpresentationofthesetwo majortopicsduringthetwosemesters,Ihavediscussedthempracticallyindependentlyofeachother.
Chapters 9 and 10,withwhichthisbookbegins,reproduceincompressedand generalizedform,essentiallyallofthemostimportantresultsthatwereobtained inthefirstpartconcerningcontinuousanddifferentiablefunctions.Thesechapters arestarredandwrittenasanappendixtoPart1.Thisappendixcontains,however, manyconceptsthatplayaroleinanyexpositionofanalysistomathematicians. Thepresenceofthesetwochaptersmakesthesecondbookformallyindependent ofthefirst,providedthereaderissufficientlywellpreparedtogetbywithoutthe numerousexamplesandintroductoryconsiderationsthat,inthefirstpart,preceded theformalismdiscussedhere.
Themainnewmaterialinthebook,whichisdevotedtotheintegralcalculusof severalvariables,beginsinChap. 11.Onewhohascompletedthefirstpartmay beginthesecondpartofthecourseatthispointwithoutanylossofcontinuityinthe ideas.
Thelanguageofdifferentialformsisexplainedandusedinthediscussionofthe theoryoflineandsurfaceintegrals.Allthebasicgeometricconceptsandanalytic constructionsthatlaterformascaleofabstractdefinitionsleadingtothegeneralized Stokes’formulaarefirstintroducedbyusingelementarymaterial.
Chapter 15 isdevotedtoasimilarsummaryexpositionoftheintegrationofdifferentialformsonmanifolds.Iregardthischapterasaverydesirableandsystematizingsupplementtowhatwasexpoundedandexplainedusingspecificobjectsin themandatoryChaps. 11–14
Thesectiononseriesandintegralsdependingonaparametergives,alongwith thetraditionalmaterial,someelementaryinformationonasymptoticseriesand
asymptoticsofintegrals(Chap. 19),since,duetoitseffectiveness,thelatterisan unquestionablyusefulpieceofanalyticmachinery.
Forconvenienceinorientation,ancillarymaterialorsectionsthatmaybeomitted onafirstreading,arestarred.
Thenumberingofthechaptersandfiguresinthisbookcontinuesthenumbering ofthefirstpart.
Biographicalinformationisgivenhereonlyforthosescholarsnotmentionedin thefirstpart.
Asbefore,fortheconvenienceofthereader,andtoshortenthetext,theendofa proofisdenotedby .Whereconvenient,definitionsareintroducedbythespecial symbols := or =: (equalitybydefinition),inwhichthecolonstandsonthesideof theobjectbeingdefined.
ContinuingthetraditionofPart1,agreatdealofattentionhasbeenpaidtoboth thelucidityandlogicalclarityofthemathematicalconstructionsthemselvesandthe demonstrationofsubstantiveapplicationsinnaturalscienceforthetheorydeveloped.
Moscow,RussiaV.Zorich 1982
10.3.4ThePartialDerivativesofaMapping...
10.4TheFinite-IncrementTheoremandSomeExamplesofItsUse..
oftheValuesofthe n thDifferential............
10.5.3SymmetryoftheHigher-OrderDifferentials
10.6Taylor’sFormulaandtheStudyofExtrema............
10.6.1Taylor’sFormulaforMappings
10.6.2MethodsofStudyingInteriorExtrema..
10.6.3SomeExamples......................
10.6.4ProblemsandExercises..................
11.1TheRiemannIntegraloveran n -DimensionalInterval
11.1.2TheLebesgueCriterionforRiemannIntegrability
11.1.4ProblemsandExercises..................
11.2TheIntegraloveraSet.......................
11.2.1AdmissibleSets......................
11.2.3TheMeasure(Volume)ofanAdmissibleSet.......
11.2.4ProblemsandExercises..................
11.3GeneralPropertiesoftheIntegral
11.3.1TheIntegralasaLinearFunctional ............
11.3.2AdditivityoftheIntegral.................
11.3.3EstimatesfortheIntegral.................
11.3.4ProblemsandExercises..................
11.4ReductionofaMultipleIntegraltoanIteratedIntegral
11.4.1Fubini’sTheorem...
11.5ChangeofVariableinaMultipleIntegral
11.5.1StatementoftheProblemandHeuristicDerivation oftheChangeofVariableFormula
11.5.2MeasurableSetsandSmoothMappings..
11.5.3TheOne-DimensionalCase
11.5.4TheCaseofanElementaryDiffeomorphismin
11.5.5CompositeMappingsandtheFormulaforChange ofVariable.........................
11.5.6AdditivityoftheIntegralandCompletionoftheProof oftheFormulaforChangeofVariableinanIntegral...
11.5.7CorollariesandGeneralizationsoftheFormula forChangeofVariableinaMultipleIntegral
11.5.8ProblemsandExercises..................
11.6ImproperMultipleIntegrals..
11.6.1BasicDefinitions......................
11.6.2TheComparisonTestforConvergenceofanImproper Integral...........................
11.6.3ChangeofVariableinanImproperIntegral
11.6.4ProblemsandExercises..................
12.1Surfacesin
12.1.1ProblemsandExercises..................
12.2OrientationofaSurface......................
12.2.1ProblemsandExercises..................
12.3TheBoundaryofaSurfaceandItsOrientation..
12.3.1SurfaceswithBoundary
12.3.2MakingtheOrientationsofaSurfaceandItsBoundary Consistent.........................
12.3.3ProblemsandExercises..................
12.4TheAreaofaSurfaceinEuclideanSpace
12.4.1ProblemsandExercises..................
12.5ElementaryFactsAboutDifferentialForms
12.5.1DifferentialForms:DefinitionandExamples
12.5.2CoordinateExpressionofaDifferentialForm......
12.5.3TheExteriorDifferentialofaForm............
12.5.4TransformationofVectorsandFormsUnderMappings.
12.5.5FormsonSurfaces...
12.5.6ProblemsandExercises..................
13LineandSurfaceIntegrals
13.1TheIntegralofaDifferentialForm................
13.1.1TheOriginalProblems,SuggestiveConsiderations, Examples..........................
13.1.2DefinitionoftheIntegralofaFormoveranOriented Surface...........................
13.2TheVolumeElement.IntegralsofFirstandSecondKind
13.2.1TheMassofaLamina...................
13.2.2TheAreaofaSurfaceastheIntegralofaForm......
13.2.3TheVolumeElement...................
13.2.4ExpressionoftheVolumeElementinCartesian Coordinates........................
13.2.5IntegralsofFirstandSecondKind
13.2.6ProblemsandExercises..................
13.3TheFundamentalIntegralFormulasofAnalysis.
13.3.1Green’sTheorem...
13.3.2TheGauss–OstrogradskiiFormula............
13.3.3Stokes’Formulain R3
13.3.4TheGeneralStokesFormula
13.3.5ProblemsandExercises..................
14.1TheDifferentialOperationsofVectorAnalysis..........
14.1.1ScalarandVectorFields
14.1.2VectorFieldsandFormsin R3
14.1.3TheDifferentialOperatorsgrad,curl,div,and ∇
14.1.4SomeDifferentialFormulasofVectorAnalysis......
14.1.5 ∗ VectorOperationsinCurvilinearCoordinates
14.1.6ProblemsandExercises..................
14.2TheIntegralFormulasofFieldTheory
14.2.1TheClassicalIntegralFormulasinVectorNotation....
14.2.2ThePhysicalInterpretationofdiv,curl,andgrad.....
14.2.3OtherIntegralFormulas..................
14.2.4ProblemsandExercises..................
14.3PotentialFields..........................
14.3.1ThePotentialofaVectorField..............
14.3.2NecessaryConditionforExistenceofaPotential
14.3.3CriterionforaFieldtobePotential............
14.3.4TopologicalStructureofaDomainandPotentials
14.3.5VectorPotential.ExactandClosedForms
14.4ExamplesofApplications.....................
14.4.1TheHeatEquation.....................
14.4.2TheEquationofContinuity................
14.4.3TheBasicEquationsoftheDynamicsofContinuous Media...........................
14.4.4TheWaveEquation....................
14.4.5ProblemsandExercises..................
15*IntegrationofDifferentialFormsonManifolds ...........
15.1ABriefReviewofLinearAlgebra
15.1.1TheAlgebraofForms.
15.1.2TheAlgebraofSkew-SymmetricForms.
15.1.3LinearMappingsofVectorSpacesandtheAdjoint MappingsoftheConjugateSpaces
15.1.4ProblemsandExercises..................
15.2Manifolds.............................
15.2.1DefinitionofaManifold
15.2.2SmoothManifoldsandSmoothMappings
15.2.3OrientationofaManifoldandItsBoundary
15.2.4PartitionsofUnityandtheRealizationofManifolds asSurfacesin R
15.2.5ProblemsandExercises..................
15.3DifferentialFormsandIntegrationonManifolds.........
15.3.1TheTangentSpacetoaManifoldataPoint
15.3.2DifferentialFormsonaManifold.............
15.3.3TheExteriorDerivative..................
15.3.4TheIntegralofaFormoveraManifold..........
15.3.5Stokes’Formula......................
15.3.6ProblemsandExercises..................
15.4ClosedandExactFormsonManifolds
15.4.1Poincaré’sTheorem..
15.4.2HomologyandCohomology
15.4.3ProblemsandExercises..................
16.1.1PointwiseConvergence
16.1.2StatementoftheFundamentalProblems.
16.1.3ConvergenceandUniformConvergenceofaFamily ofFunctionsDependingonaParameter..
16.1.4TheCauchyCriterionforUniformConvergence
16.1.5ProblemsandExercises..................
16.2UniformConvergenceofSeriesofFunctions...
16.2.1BasicDefinitionsandaTestforUniformConvergence ofaSeries.........................
16.2.2TheWeierstrass M -TestforUniformConvergence ofaSeries.........................
16.2.3TheAbel–DirichletTest..................
16.2.4ProblemsandExercises..................
16.3FunctionalPropertiesofaLimitFunction
16.3.1SpecificsoftheProblem
16.3.2ConditionsforTwoLimitingPassagestoCommute...
16.3.5DifferentiationandPassagetotheLimit.........
16.3.6ProblemsandExercises..................
16.4*CompactandDenseSubsetsoftheSpaceofContinuous
16.4.1TheArzelà–AscoliTheorem
17.1ProperIntegralsDependingonaParameter
17.1.1TheConceptofanIntegralDependingonaParameter..
17.2.1UniformConvergenceofanImproperIntegral withRespecttoaParameter
17.2.2LimitingPassageUndertheSignofanImproperIntegral andContinuityofanImproperIntegralDepending onaParameter.......................
17.2.3DifferentiationofanImproperIntegralwithRespect toaParameter.......................
17.2.4IntegrationofanImproperIntegralwithRespect toaParameter.......................
17.2.5ProblemsandExercises..................
17.4ConvolutionofFunctionsandElementaryFacts
17.4.1ConvolutioninPhysicalProblems(Introductory Considerations)......................
17.4.2GeneralPropertiesofConvolution
17.4.3ApproximateIdentitiesandtheWeierstrass ApproximationTheorem
17.4.4*ElementaryConceptsInvolvingDistributions
17.4.5ProblemsandExercises..................
17.5MultipleIntegralsDependingonaParameter... ........
17.5.1ProperMultipleIntegralsDependingonaParameter...
17.5.2ImproperMultipleIntegralsDependingonaParameter. 472
17.5.3ImproperIntegralswithaVariableSingularity
17.5.4*Convolution,theFundamentalSolution,and GeneralizedFunctionsintheMultidimensionalCase...
17.5.5ProblemsandExercises.................. 488
18FourierSeriesandtheFourierTransform
18.1BasicGeneralConceptsConnectedwithFourierSeries
18.1.1OrthogonalSystemsofFunctions
18.1.2FourierCoefficientsandFourierSeries..........
18.1.3*AnImportantSourceofOrthogonalSystems ofFunctionsinAnalysis .................. 510
18.1.4ProblemsandExercises..................
18.2TrigonometricFourierSeries. ..................
18.2.1BasicTypesofConvergenceofClassicalFourierSeries. 520
18.2.2InvestigationofPointwiseConvergence ofaTrigonometricFourierSeries ............. 524
18.2.3SmoothnessofaFunctionandtheRateofDecrease oftheFourierCoefficients................. 534
18.2.4CompletenessoftheTrigonometricSystem 539
18.2.5ProblemsandExercises.................. 545
18.3TheFourierTransform....................... 553
18.3.1RepresentationofaFunctionbyMeansofaFourier Integral........................... 553
18.3.2TheConnectionoftheDifferentialandAsymptotic PropertiesofaFunctionandItsFourierTransform 566
18.3.3TheMainStructuralPropertiesoftheFourierTransform. 569
18.3.4ExamplesofApplications................. 574
18.3.5ProblemsandExercises.................. 579
19AsymptoticExpansions
19.1AsymptoticFormulasandAsymptoticSeries........... 589
19.1.1BasicDefinitions...................... 589
19.1.2GeneralFactsAboutAsymptoticSeries.. ........ 594
19.1.3AsymptoticPowerSeries................. 598
19.1.4ProblemsandExercises.................. 600
19.2TheAsymptoticsofIntegrals(Laplace’sMethod) ........ 603
19.2.1TheIdeaofLaplace’sMethod
19.2.2TheLocalizationPrincipleforaLaplaceIntegral.....
19.2.3CanonicalIntegralsandTheirAsymptotics
19.2.4ThePrincipalTermoftheAsymptoticsofaLaplace Integral...........................
19.2.5*AsymptoticExpansionsofLaplaceIntegrals
19.2.6ProblemsandExercises..................
A.1.1TheSmallBugontheRubberRope
A.1.2IntegralandEstimationofSums..............
A.1.3FromMonkeystoDoctorsofScienceAltogetherin106
A.2TheExponentialFunction...
A.2.1PowerSeriesExpansionoftheFunctionsexp,sin,cos.. 648
A.2.2ExittotheComplexDomainandEuler’sFormula....
A.2.3TheExponentialFunctionasaLimit...
A.2.4MultiplicationofSeriesandtheBasicProperty oftheExponentialFunction 649
A.2.5ExponentialofaMatrixandtheRoleofCommutativity. 650
A.2.6ExponentialofOperatorsandTaylor’sFormula ...... 650
A.3Newton’sBinomial........................
A.3.1ExpansioninPowerSeriesoftheFunction (1 + x)α ... 651
A.3.2IntegrationofaSeriesandExpansionofln(1 + x) .... 651
A.3.3ExpansionoftheFunctions (1 + x 2 ) 1 andarctan x ... 651
A.3.4Expansionof (1 + x) 1 andComputingCuriosities... 652
A.4SolutionofDifferentialEquations................. 652
A.4.1MethodofUndeterminedCoefficients.. ........ 652
A.4.2UseoftheExponentialFunction 652
A.5TheGeneralIdeaAboutApproximationandExpansion 653
A.5.1TheMeaningofaPositionalNumberSystem.Irrational Numbers.......................... 653
A.5.2ExpansionofaVectorinaBasisandSomeAnalogies withSeries.........................
A.5.3Distance
AppendixBChangeofVariablesinMultipleIntegrals(Deduction andFirstDiscussionoftheChangeofVariablesFormula) 655
B.1FormulationoftheProblemandaHeuristicDerivation oftheChangeofVariablesFormula ................ 655
B.2SomePropertiesofSmoothMappingsandDiffeomorphisms.. 656
B.3RelationBetweentheMeasuresoftheImageandthePre-image UnderDiffeomorphisms ...................... 657
B.4SomeExamples,Remarks,andGeneralizations.. ........ 659
AppendixCMultidimensionalGeometryandFunctionsofaVery LargeNumberofVariables(ConcentrationofMeasuresandLaws ofLargeNumbers) ...........................
C.1AnObservation..........................
C.2SphereandRandomVectors..
C.3MultidimensionalSphere,LawofLargeNumbers,andCentral LimitTheorem ........................... 664
C.4MultidimensionalIntervals(MultidimensionalCubes) ...... 666
C.5GaussianMeasuresandTheirConcentration...
C.6ALittleBitMoreAbouttheMultidimensionalCube
C.7TheCodingofaSignalinaChannelwithNoise. ........ 668
AppendixDOperatorsofFieldTheoryinCurvilinearCoordinates
Introduction...
D.1RemindersofAlgebraandGeometry 671
D.1.1BilinearFormsandTheirCoordinateRepresentation... 671
D.1.2CorrespondenceBetweenFormsandVectors 672
D.1.3CurvilinearCoordinatesandMetric 673
D.2Operatorsgrad,curl,divinCurvilinearCoordinates 674
D.2.1DifferentialFormsandOperatorsgrad,curl,div..... 674
D.2.2GradientofaFunctionandItsCoordinateRepresentation 675
D.2.3DivergenceandItsCoordinateRepresentation 676
D.2.4CurlofaVectorFieldandItsCoordinateRepresentation. 680
AppendixEModernFormulaofNewton–LeibnizandtheUnity ofMathematics(FinalSurvey) .................... 683
E.1Reminders. ............................ 683
E.1.1Differential,DifferentialForm,andtheGeneralStokes’s Formula.......................... 683
E.1.2Manifolds,Chains,andtheBoundaryOperator ...... 686
E.2Pairing............................... 688
E.2.1TheIntegralasaBilinearFunctionandGeneralStokes’s Formula.......................... 688
E.2.2EquivalenceRelations(HomologyandCohomology).. 689
Chapter9
*ContinuousMappings(GeneralTheory)
Inthischapterweshallgeneralizethepropertiesofcontinuousmappingsestablished earlierfornumerical-valuedfunctionsandmappingsofthetype f : Rm → Rn and discussthemfromaunifiedpointofview.Intheprocessweshallintroduceanumber ofsimple,yetimportantconceptsthatareusedeverywhereinmathematics.
9.1MetricSpaces
9.1.1DefinitionandExamples
Definition1 Aset X issaidtobeendowedwitha metric or ametricspacestructure ortobea metricspace ifafunction
d : X × X → R
isexhibitedsatisfyingthefollowingconditions:
a) d(x1 ,x2 ) = 0 ⇔ x1 = x2 ,
b) d(x1 ,x2 ) = d(x2 ,x1 ) (symmetry),
c) d(x1 ,x3 ) ≤ d(x1 ,x2 ) + d(x2 ,x3 ) (thetriangleinequality), where x1 ,x2 ,x3 arearbitraryelementsof X .
Inthatcase,thefunction(9.1)iscalleda metric or distance on X Thusa metricspace isapair (X,d) consistingofaset X andametricdefinedon it.
Inaccordancewithgeometricterminologytheelementsof X arecalled points Weremarkthatifweset x3 = x1 inthetriangleinequalityandtakeaccountof conditionsa)andb)inthedefinitionofametric,wefindthat 0 ≤ d(x1 ,x2 ), thatis,adistancesatisfyingaxiomsa),b),andc)isnonnegative.
©Springer-VerlagBerlinHeidelberg2016 V.A.Zorich, MathematicalAnalysisII,Universitext, DOI 10.1007/978-3-662-48993-2_1
(9.1)
Letusnowconsidersomeexamples.
Example1 Theset R ofrealnumbersbecomesametricspaceifweset d(x1 ,x2 ) = |x2 x1 | foranytwonumbers x1 and x2 ,aswehavealwaysdone.
Example2 Othermetricscanalsobeintroducedon R.Atrivialmetric,forexample, isthediscretemetricinwhichthedistancebetweenanytwodistinctpointsis1.
Thefollowingmetricon R ismuchmoresubstantive.Let x → f(x) beanonnegativefunctiondefinedfor x ≥ 0andvanishingfor x = 0.Ifthisfunctionisstrictly convexupward,then,setting
,x
forpoints x1 ,x2 ∈ R,weobtainametricon R. Axiomsa)andb)obviouslyholdhere,andthetriangleinequalityfollowsfrom theeasilyverifiedfactthat f isstrictlymonotonicandsatisfiesthefollowinginequalitiesfor0 <a<b : f(a + b) f(b)<f(a) f(0) = f(a).
Inparticular,onecouldset d(x1 ,x2 ) = √|x1 x2 | or d(x1 ,x2 ) = |
| 1+|x1 x
| .In thelattercasethedistancebetweenanytwopointsofthelineislessthan1.
Example3 Besidesthetraditionaldistance
betweenpoints x1 = (x 1 1 ,...,x n 1 ) and x2 = (x 1 2 ,...,x n 2 ) in Rn ,onecanalsointroducethedistance
where p ≥ 1.Thevalidityofthetriangleinequalityforthefunction(9.4)follows fromMinkowski’sinequality(seeSect.5.4.2).
Example4 Whenweencounterawordwithincorrectletterswhilereadingatext, wecanreconstructthewordwithouttoomuchtroublebycorrectingtheerrors, providedthenumberoferrorsisnottoolarge.However,correctingtheerrorand obtainingthewordisanoperationthatissometimesambiguous.Forthatreason, otherconditionsbeingequal,onemustgivepreferencetotheinterpretationofthe incorrecttextthatrequiresthefewestcorrections.Accordingly,incodingtheory
9.1MetricSpaces3
themetric(9.4)with p = 1isusedonthesetofallfinitesequencesoflength n consistingofzerosandones.
Geometricallythesetofsuchsequencescanbeinterpretedasthesetofverticesoftheunitcube I ={x ∈ Rn | 0 ≤ x i ≤ 1,i = 1,...,n} in Rn .Thedistance betweentwoverticesisthenumberofinterchangesofzerosandonesneededto obtainthecoordinatesofonevertexfromtheother.Eachsuchinterchangerepresentsapassagealongoneedgeofthecube.Thusthisdistanceistheshortestpath alongtheedgesofthecubefromoneoftheverticesunderconsiderationtothe other.
Example5 Incomparingtheresultsoftwoseriesof n measurementsofthesame quantitythemetricmostcommonlyusedis(9.4)with p = 2.Thedistancebetween pointsinthismetricisusuallycalledtheir mean-squaredeviation
Example6 Asonecaneasilysee,ifwepasstothelimitin(9.4)as p →+∞ ,we obtainthefollowingmetricin Rn :
Example7 Theset C [a,b ] offunctionsthatarecontinuousonaclosedinterval becomesametricspaceifwedefinethedistancebetweentwofunctions f and g to be
Axiomsa)andb)forametricobviouslyhold,andthetriangleinequalityfollows fromtherelations
f(x) h(x) ≤ f(x) g(x) + g(x) h(x) ≤ d(f,g) + d(g,h), thatis, d(f,h) = max a ≤x ≤b f(x) h(x) ≤ d(f,g) + d(g,h).
Themetric(9.6)–theso-called uniform or Chebyshev metricin C [a,b ] –is usedwhenwewishtoreplaceonefunctionbyanother(forexample,apolynomial) fromwhichitispossibletocomputethevaluesofthefirstwitharequireddegreeof precisionatanypoint x ∈[a,b ].Thequantity d(f,g) ispreciselyacharacterization oftheprecisionofsuchanapproximatecomputation.
Themetric(9.6)closelyresemblesthemetric(9.5)in Rn .
Example8 Likethemetric(9.4),for p ≥ 1wecanintroducein C [a,b ] themetric
=
49*ContinuousMappings(GeneralTheory)
ItfollowsfromMinkowski’sinequalityforintegrals,whichcanbeobtainedfrom Minkowski’sinequalityfortheRiemannsumsbypassingtothelimit,thatthisis indeedametricfor p ≥ 1.
Thefollowingspecialcasesofthemetric(9.7)areespeciallyimportant: p = 1, whichistheintegralmetric; p = 2,themetricofmean-squaredeviation;and p = +∞ ,theuniformmetric.
Thespace C [a,b ] endowedwiththemetric(9.7)isoftendenoted Cp [a,b ].One canverifythat C∞ [a,b ] isthespace C [a,b ] endowedwiththemetric(9.6).
Example9 Themetric(9.7)couldalsohavebeenusedontheset R[a,b ] of Riemann-integrablefunctionsontheclosedinterval [a,b ].However,sincetheintegraloftheabsolutevalueofthedifferenceoftwofunctionsmayvanishevenwhen thetwofunctionsarenotidenticallyequal,axioma)willnotholdinthiscase.Nevertheless,weknowthattheintegralofanonnegativefunction ϕ ∈ R[a,b ] equals zeroifandonlyif ϕ(x) = 0atalmostallpointsoftheclosedinterval [a,b ].
Therefore,ifwepartition R[a,b ] intoequivalenceclassesoffunctions,regarding twofunctionsin R[a,b ] asequivalentiftheydifferonatmostasetofmeasurezero, thentherelation(9.7)reallydoesdefineametricontheset R[a,b ] ofsuchequivalenceclasses.Theset R[a,b ] endowedwiththismetricwillbedenoted Rp [a,b ] andsometimessimplyby Rp [a,b ].
Example10 Intheset C (k) [a,b ] offunctionsdefinedon [a,b ] andhavingcontinuousderivativesuptoorder k inclusiveonecandefinethefollowingmetric:
d(f,g) = max{M0 ,...,Mk },
(i) (x) g (i) (x) ,i = 0, 1,...,k.
Usingthefactthat(9.6)isametric,onecaneasilyverifythat(9.8)isalsoa metric.
Assumeforexamplethat f isthecoordinateofamovingpointconsidered asafunctionoftime.Ifarestrictionisplacedontheallowableregionwhere thepointcanbeduringthetimeinterval [a,b ] andtheparticleisnotallowed toexceedacertainspeed,and,inaddition,wewishtohavesomeassurance thattheaccelerationscannotexceedacertainlevel,itisnaturaltoconsiderthe set {maxa ≤x ≤b |f(x)|, maxa ≤x ≤b |f (x)|, maxa ≤x ≤b |f (x)|} forafunction f ∈ C (2) [a,b ] andusingthesecharacteristics,toregardtwomotions f and g asclose togetherifthequantity(9.8)forthemissmall.
Theseexamplesshowthatagivensetcanbemetrizedinvariousways.The choiceofthemetrictobeintroducedisusuallycontrolledbythestatementofthe problem.Atpresentweshallbeinterestedinthemostgeneralpropertiesofmetric spaces,thepropertiesthatareinherentinallofthem.
9.1.2OpenandClosedSubsetsofaMetricSpace
Let (X,d) beametricspace.Inthegeneralcase,aswasdoneforthecase X = Rn inSect.7.1,onecanalsointroducetheconceptofaballwithcenteratagivenpoint, openset,closedset,neighborhoodofapoint,limitpointofaset,andsoforth. Letusnowrecalltheseconcepts,whicharebasicforwhatistofollow.
Definition2 For δ> 0and a ∈ X theset
B(a,δ) = x ∈ X | d(a,x)<δ
iscalledthe ballwithcenter a ∈ X ofradius δ orthe δ -neighborhoodofthepoint a
Thisnameisaconvenientoneinageneralmetricspace,butitmustnotbeidentifiedwiththetraditionalgeometricimagewearefamiliarwithin R3 .
Example11 Theunitballin C [a,b ] withcenteratthefunctionthatisidentically0 on [a,b ] consistsofthefunctionsthatarecontinuousontheclosedinterval [a,b ] andwhoseabsolutevaluesarelessthan1onthatinterval.
Example12 Let X betheunitsquarein R2 forwhichthedistancebetweentwo pointsisdefinedtobethedistancebetweenthosesamepointsin R2 .Then X isa metricspace,whilethesquare X consideredasametricspaceinitsownrightcan beregardedastheballofanyradius ρ ≥ √2/2aboutitscenter.
Itisclearthatinthiswayonecouldconstructballsofverypeculiarshape.Hence thetermballshouldnotbeunderstoodtooliterally.
Definition3 Aset G ⊂ X is openinthemetricspace (X,d) ifforeachpoint x ∈ G thereexistsaball B(x,δ) suchthat B(x,δ) ⊂ G .
Itobviouslyfollowsfromthisdefinitionthat X itselfisanopensetin (X,d) Theemptyset ∅ isalsoopen.Bythesamereasoningasinthecaseof Rn onecan provethataball B(a,r) anditsexterior {x ∈ X : d(a,x)>r } areopensets.(See Examples3and4ofSect.7.1.)
Definition4 Aset F ⊂ X is closedin (X,d) ifitscomplement X \F isopenin (X,d) .
Inparticular,weconcludefromthisdefinitionthatthe closedball
B(a,r) := x ∈ X | d(a,x) ≤ r
isaclosedsetinametricspace (X,d) .
Thefollowingpropositionholdsforopenandclosedsetsinametricspace (X,d) .
Proposition1 a) Theunion α ∈A Gα ofthesetsinanysystem {Gα ,α ∈ A} ofsets Gα thatareopenin X isanopensetin X
b) Theintersection n i =1 Gi ofanyfinitenumberofsetsthatareopenin X isan opensetin X
a ) Theintersection α ∈A Fα ofthesetsinanysystem {Fα ,α ∈ A} ofsets Fα thatareclosedin X isaclosedsetin X .
b ) Theunion n i =1 Fi ofanyfinitenumberofsetsthatareclosedin X isaclosed setin X .
TheproofofProposition 1 isaverbatimrepetitionoftheproofofthecorrespondingpropositionforopenandclosedsetsin Rn ,andweomitit.(SeeProposition1in Sect.7.1.)
Definition5 Anopensetin X containingthepoint x ∈ X iscalleda neighborhood ofthepoint x in X .
Definition6 Relativetoaset E ⊂ X ,apoint x ∈ X iscalled aninteriorpointof E ifsomeneighborhoodofitiscontainedin X , anexteriorpointof E ifsomeneighborhoodofitiscontainedinthecomplement of E in X , aboundarypointof E ifitisneitherinteriornorexteriorto E (thatis,every neighborhoodofthepointcontainsbothapointbelongingto E andapointnot belongingto E ).
Example13 Allpointsofaball B(a,r) areinteriortoit,andtheset CX B(a,r) = X \B(a,r) consistsofthepointsexteriortotheball B(a,r) Inthecaseof Rn withthestandardmetric d the sphere S(a,r) :={x ∈ Rn | d(a,x) = r ≥ 0} isthesetofboundarypointsoftheball B(a,r) 1
Definition7 Apoint a ∈ X isa limitpoint oftheset E ⊂ X iftheset E ∩ O(a) is infiniteforeveryneighborhood O(a) ofthepoint.
Definition8 Theunionoftheset E andthesetofallitslimitpointsiscalledthe closure oftheset E in X Asbefore,theclosureofaset E ⊂ X willbedenoted E
Proposition2 Aset F ⊂ X isclosedin X ifandonlyifitcontainsallitslimit points.
Thus (F isclosedin X) ⇐⇒ (F = F in X).
1 InconnectionwithExample 13 seealsoProblem 2 attheendofthissection.
Weomittheproof,sinceitrepeatstheproofoftheanalogouspropositionforthe case X = Rn discussedinSect.7.1.
9.1.3SubspacesofaMetricSpace
If (X,d) isametricspaceand E isasubsetof X ,then,settingthedistancebetween twopoints x1 and x2 of E equalto d(x1 ,x2 ) ,thatis,thedistancebetweenthem in X ,weobtainthemetricspace (E,d),whichiscustomarilycalleda subspace of theoriginalspace (X,d) . Thusweadoptthefollowingdefinition.
Definition9 Ametricspace (X1 ,d1 ) isa subspaceofthemetricspace (X,d) if X1 ⊂ X andtheequality d1 (a,b) = d(a,b) holdsforanypairofpoints a,b in X1
Sincetheball B1 (a,r) ={x ∈ X1 | d1 (a,x)<r } inasubspace (X1 ,d1 ) ofthe metricspace (X,d) isobviouslytheintersection
B1 (a,r) = X1 ∩ B(a,r)
oftheset X1 ⊂ X withtheball B(a,r) in X ,itfollowsthateveryopensetin X1 hastheform
G1 = X1 ∩ G,
where G isanopensetin X ,andeveryclosedset F1 in X1 hastheform
F1 = X1 ∩ F,
where F isaclosedsetin X .
Itfollowsfromwhathasjustbeensaidthatthepropertiesofasetinametric spaceofbeingopenorclosedarerelativepropertiesanddependontheambient space.
Example14 Theopeninterval |x | < 1, y = 0ofthe x -axisintheplane R2 withthe standardmetricin R2 isametricspace (X1 ,d1 ) ,which,likeanymetricspace,is closedasasubsetofitself,sinceitcontainsallitslimitpointsin X1 .Atthesame time,itisobviouslynotclosedin R2 = X .
Thissameexampleshowsthatopennessisalsoarelativeconcept.
Example15 Theset C [a,b ] ofcontinuousfunctionsontheclosedinterval [a,b ] withthemetric(9.7)isasubspaceofthemetricspace Rp [a,b ].However,ifwe considerthemetric(9.6)on C [a,b ] ratherthan(9.7),thisisnolongertrue.
9.1.4TheDirectProductofMetricSpaces
If (X1 ,d1 ) and (X2 ,d2 ) aretwometricspaces,onecanintroduceametric d onthe directproduct X1 × X2 .Thecommonestmethodsofintroducingametricin X1 × X2 arethefollowing.If (x1 ,x2 ) ∈
and (x1 ,x2 ) ∈
,onemayset
or
or
Itiseasytoseethatweobtainametricon X1 × X2 inallofthesecases.
Definition10 if (X1 ,d1 ) and (X2 ,d2 ) aretwometricspaces,thespace (X1 × X2 ,d) ,where d isametricon X1 × X2 introducedbyanyofthemethodsjust indicated,willbecalledthe directproduct oftheoriginalmetricspaces.
Example16 Thespace R2 canberegardedasthedirectproductoftwocopiesof themetricspace R withitsstandardmetric,and R3 isthedirectproduct R2 × R1 of thespaces R2 and R1 = R.
9.1.5ProblemsandExercises
1. a)ExtendingExample 2,showthatif f : R+ → R+ isacontinuousfunction thatisstrictlyconvexupwardandsatisfies f(0) = 0,while (X,d) isametricspace, thenonecanintroduceanewmetric df on X bysetting df (x1 ,x2 ) = f(d(x1 ,x2 )) .
b)Showthatonanymetricspace (X,d) onecanintroduceametric d (x1 ,x2 ) = d(x1 ,x2 ) 1+d(x1 ,x2 ) inwhichthedistancebetweenthepointswillbelessthan1.
2. Let (X,d) beametricspacewiththetrivial(discrete)metricshowninExample 2,andlet a ∈ X .Forthiscase,whatarethesets B(a, 1/2) , B(a, 1) , B(a, 1) , B(a, 1) , B(a, 3/2) ,andwhatarethesets {x ∈ X | d(a,x) = 1/2} , {x ∈ X | d(a,x) = 1} , B(a, 1)\B(a, 1) , B(a, 1)\B(a, 1) ?
3. a)Isittruethattheunionofanyfamilyofclosedsetsisaclosedset?
b)Iseveryboundarypointofasetalimitpointofthatset?
c)Isittruethatinanyneighborhoodofaboundarypointofasettherearepoints inboththeinteriorandexteriorofthatset?
d)Showthatthesetofboundarypointsofanysetisaclosedset.
4. a)Provethatif (Y,dY ) isasubspaceofthemetricspace (X,dX ) ,thenforany open(resp.closed)set GY (resp. FY )in Y thereisanopen(resp.closed)set GX (resp. FX )in X suchthat GY = Y ∩ GX ,(resp. FY = Y ∩ FX ).
b)Verifythatiftheopensets GY and GY in Y donotintersect,thenthecorrespondingsets GX and GX in X canbechosensothattheyalsohavenopointsin common.
5. Havingametric d onaset X ,onemayattempttodefinethedistance d(A,B) betweensets A ⊂ X and B ⊂ X asfollows:
= inf
a)Giveanexampleofametricspaceandtwononintersectingsubsetsofit A and B forwhich d(A,B) = 0.
b)Show,followingHausdorff,thatonthesetofclosedsetsofametricspace (X,d) onecanintroducethe Hausdorffmetric D byassumingthatfor A ⊂ X and B ⊂ X
9.2TopologicalSpaces
Forquestionsconnectedwiththeconceptofthelimitofafunctionoramapping, whatisessentialinmanycasesisnotthepresenceofanyparticularmetriconthe space,butratherthepossibilityofsayingwhataneighborhoodofapointis.To convinceoneselfofthatitsufficestorecallthattheverydefinitionofalimitorthe definitionofcontinuitycanbestatedintermsofneighborhoods.Topologicalspaces arethemathematicalobjectsonwhichtheoperationofpassagetothelimitandthe conceptofcontinuitycanbestudiedinmaximumgenerality.
9.2.1BasicDefinitions
Definition1 Aset X issaidtobeendowedwiththestructureofa topologicalspace ora topology orissaidtobea topologicalspace ifasystem τ ofsubsetsof X is exhibited(called opensetsin X )possessingthefollowingproperties:
a) ∅ ∈ τ ; X ∈ τ .
b) (∀α ∈ A(τα ∈ τ)) =⇒ α ∈A τα ∈ τ .
c) (τi ∈ τ ; i = 1,...,n) =⇒ n i =1 τi ∈ τ .
Thus,atopologicalspaceisapair (X,τ) consistingofaset X andasystem τ ofdistinguishedsubsetsofthesethavingthepropertiesthat τ containstheempty setandthewholeset X ,theunionofanynumberofsetsof τ isasetof τ ,andthe intersectionofanyfinitenumberofsetsof τ isasetof τ Asonecansee,intheaxiomsystema),b),c)foratopologicalspacewehave postulatedpreciselythepropertiesofopensetsthatwealreadyprovedinthecaseof
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In 1804, Aaron Burr, presenting himself as a candidate for Governor of New York, but Hamilton opposed his election expressing the opinion that “Burr was a dangerous man and unfit to be trusted with power.” The election of Gen. Lewis blasted the ambitious projects of Burr, who insolently demanded an explanation of Hamilton, and finally challenged him, Hamilton accepted the challenge, was mortally wounded at Weehawken, and died July 12, 1804. His death was profoundly lamented throughout the country.
N His eldest son had been killed in a duel by a political adversary about 1802 Mr Hamilton was the principal author of the Federalist, and the real father of our financial system Immediately after adopting the constitution, he strongly advocated the establishment of a Mint, so that the New World would not be dependant on the Old for a circulating medium.
DIRECTORS OF THE MINT.
DAVID RITTENHOUSE, F D M .
Entering the Cabinet, the portraits of the different Directors attract attention. That of David Rittenhouse is the copy of a painting by Charles Willson Peale. Mr. Rittenhouse was appointed by Washington, April 14, 1792, and remained in charge of the Mint until June, 1795, when his declining health compelled him to resign.
At an early age he indicated mechanical talent of a high order in the construction of a clock, and his studies from that time were principally mathematical. His genius soon attracted attention, and he was appointed by the colonial governor a surveyor, and in that capacity determined the famous Mason and Dixon line. He succeeded Benjamin Franklin as President of the American Philosophical Society. Mr. Barber, late Engraver of the Mint, executed a bronze medal of Dr. Rittenhouse. Possibly, excepting Duvivier’s head of Washington after Houdon, it cannot be surpassed in the Cabinet. The engraver had a very fine subject, and treated it in the highest style of art. On the obverse is “David Rittenhouse,” with date of birth and death. On the reverse, inscription, “He belonged to the whole human race.”—“Wm. Barber.” This beautiful memento is highly prized.
HENRY WILLIAM DESAUSSURE, S D M .
The portrait of Henry William Desaussure, now in the cabinet, was painted by Samuel Du Bois, from a daguerreotype taken from a family picture. This Director was distinguished for his legal ability, as
well as his strict integrity He entered upon his duties with a protest, as he claimed no knowledge of the requirements of the position, having long been a practicing lawyer; but he was reassured by Alex. Hamilton, then Secretary of the Treasury, and proved himself a fine officer for the short term of his service. He was appointed by Washington, July 8, 1795, but resigned in the following October. Washington not only expressed regret at losing so valuable an officer, but consulted him as to the selection of a successor.
ELIAS BOUDINOT, T D M ,
was appointed October 28, 1795, and remained in office eleven years. In the summer and autumn of 1797 and the two following years, and also of 1802 and 1803, the Mint was closed on account of the ravages of the yellow fever. Mr. Boudinot resigned in 1805, and devoted the remainder of his life to benevolent and literary pursuits. He died on the 24th of October, 1821, at the advanced age of eightytwo. The fine portrait of this venerable Director seen in the Cabinet was presented by a relative, and is a good copy of a painting by Waldo and Jewett.
ROBERT PATTERSON, LL.D.,
F D M ,
was appointed by President Jefferson, January 17, 1806. He was a native of Ireland, distinguished for his acquirements and ability He held the office of Director for an exceptionally long term of service. His portrait, which hangs in the Cabinet, is a copy of a fine original by Rembrandt Peale.
SAMUEL MOORE, M. D.,
F D M ,
was appointed by President James Monroe, July 15, 1824. He was a native of New Jersey, and the son of a distinguished Revolutionary officer. He was one of the first graduates of the Penn University, in 1791, and was afterwards a tutor in that institution. During his directorship the Mint was removed to the present building. His portrait was painted from life by B. Samuel Du Bois, now in the Cabinet.
ROBERT MASKELL PATTERSON, M. D., S D M ,
son of a former Director, was appointed by President Andrew Jackson, May 26, 1835. His term of office was marked by an entire revolution in the coinage, and the ready acceptance of those improvements which followed so rapidly upon the introduction of steam. Dr. Patterson possessed the advantage of foreign travel; and having become familiar with the discoveries which had been adopted in the French Mint, he inaugurated and perfected them, also introducing improvements, which are still in use, in the machinery of the Mint. His portrait is in the Cabinet.
GEORGE N. ECKERT, M. D., S D M ,
was appointed by President Fillmore, July 1, 1851. He served nearly two years, and, resigning, was followed by
THOMAS M. PETTIT, E D M ,
who was appointed by President Pierce, April 4, 1853. He died a few weeks after his appointment. No portrait of him in the Cabinet. He was succeeded by
HON. JAMES ROSS SNOWDEN, LL.D., N D M .
Mr. Snowden, who was appointed by President Pierce, June 3, 1853, was formerly a member of the State Legislature, and served two terms as Speaker; was afterwards elected for two terms as State Treasurer. During his official term the building was made fire-proof, the large collection of minerals was added, and nickel was first coined.
Mr. Snowden has placed the numismatic world under many obligations, by directing the publication of two valuable quarto volumes,—one of them a description of the coins in the Cabinet, under the title of “The Mint Manual of Coins of all Nations,” the other “The Medallic Memorials of Washington,” being mainly a description of a special collection made by himself. In the preface to the former work he gives due credit to the literary labors of Mr. George Bull, then Curator, and also to a reprint of the account of the ancient collection, by Mr. Du Bois, who also furnished other valuable material. These books are valuable as authority, and by reason of the national character of the last mentioned.
JAMES POLLOCK, A.M., LL.D.,
T D F S ,
was appointed by Abraham Lincoln in 1861, and was re-appointed by President Grant to succeed Dr. Linderman in 1869 to 1873. Born in Pennsylvania in 1810; graduated at Princeton College, New Jersey, in 1831, and commenced the practice of the law in 1833; he served in Congress three terms; was elected Governor of Pennsylvania in 1854, and in 1860 was a peace delegate to Washington from his State to counsel with representatives from different parts of the Union as to the possibility of amicably adjusting our unhappy national troubles. His portrait, by Winner, hangs in the eastern section of the Cabinet.[19]
HON. HENRY RICHARD LINDERMAN, M. D.,
,
was the eldest son of John Jordan Linderman, M. D., and Rachel Brodhead. He was born in Pike county, Pennsylvania, the 25th of December, 1825. The elder Dr. Linderman was one of the most noted physicians in northeastern Pennsylvania, and practiced medicine for nearly half a century in the valley of the Delaware, in this State, and New Jersey. He was a graduate of the College of Physicians and Surgeons, of New York, where he had studied under the famous Dr. Valentine Mott. Dr. Linderman’s grandfather, Jacob von Linderman, came to this country during the disturbed period of the Austrian War of Succession, during the first half of the last century, and settled in Orange county, where he purchased a tract of land. The property is still in the possession of the family. Jacob von Linderman was the cadet of an ancient and honorable family of Saxony, which had been distinguished for two centuries in the law and medicine, several of his ancestors having been counsellors and physicians to the Elector. He was a descendant of the same family as Margaretta Linderman, the mother of the great Reformer, Martin Luther. Of this paternal stock, Dr. Henry R. Linderman was, by his mother, a nephew of the late Hon. Richard Brodhead, Senator of the United States from Pennsylvania; grandson of Richard Brodhead, one of the Judges of Pike county, and great-grandson of Garrett Brodhead, an officer of the Revolution, and a great-nephew of Luke Brodhead, a Captain in Col. Miles’ Regiment, and of Daniel Brodhead, Colonel of the 8th Pennsylvania Regiment of the Continental Line; the latter was afterwards a Brigadier-General, was one of the original members of the Cincinnati of this State, and Surveyor-General of the Commonwealth when the war closed. His only son Daniel was a First Lieutenant in Colonel Shee’s Battalion, was taken prisoner by the British, and died after two years’ captivity. General Brodhead married Governor Mifflin’s widow, and died in Milford, Pike county, in 1803. The nephew of these three brothers, Charles Wessel Brodhead, of New York, was also in the
Revolutionary army, a Captain of Grenadiers. They all descended from Daniel Brodhead, a Captain of King Charles II.’s Grenadiers, who had a command in Nichol’s expedition, which captured New York from the Dutch in 1664. Captain Brodhead was of the family of that name in Yorkshire, which terminated in England so recently as 1840 in the person of Sir Henry T. L. Brodhead, baronet.
Dr. Henry R. Linderman, after receiving an academic education, entered the New York College of Physicians and Surgeons. When barely of age he graduated, returned to Pike county and began practice with his father, and earned a reputation as a skillful and rising physician.
In 1855 his uncle, Richard Brodhead (United States Senator), procured his appointment as chief clerk of the Philadelphia Mint. He held this position until 1864, when he resigned and engaged in business as a banker and broker in Philadelphia. In 1867 he was appointed Director of the Mint by President Johnson. In 1869 he resigned. In 1870 he was a commissioner of the Government to the Pacific coast to investigate the San Francisco and Carson Mints, and to adjust some intricate bullion questions. In 1871 he was a commissioner to Europe, to examine the coinage systems of the Great Powers. In 1872 he was a commissioner, with the late Dr. Robert E. Rogers, of the University of Pennsylvania, for fitting up the Government refinery at the San Francisco Mint. In the same year he wrote an elaborate report on the condition of the gold and silver market of the world. “In this report he called attention to the disadvantages arising from the computation and quotation of exchange with Great Britain on the old and complicated Colonial basis, and from the undervaluation of foreign coins in computing the value of foreign invoices and levying and collecting duties at the United States Custom Houses.” He was the author of the Act of March 9th, 1873, which corrected the defects above referred to. His predictions in this report on the decline in the value of silver as compared to gold were fulfilled to the letter.
He was thoroughly familiar with the practice, science, and finance of the Coinage Department of the Government, and about this time he wrote the Coinage Act of 1873, and secured its passage through
Congress. General Grant, then President, considered him as the fittest man to organize the new Bureau, and, though a Democrat, appointed him first Director under the new Act; the Director being at the head of all the Mints and Assay Offices in the United States.
For the remainder of his life until his last illness, which began in the fall of 1878, he worked incessantly. Under his hands the Bureau of the Mints and the entire Coinage and Assay service were shaped in their present form. Much is due to his official subordinates, but his was the master mind, his the skillful and methodical direction, the studious and laborious devotion to the duties and obligations of his high position at the head of the Coinage Department of this great nation, which have given the United States the best coinage system in the world. It was Dr. Linderman who projected the “trade dollar,” solely for commerce, and not intended to enter into circulation here. It was a successful means of finding a market for our great surplus of silver, which Dr. Linderman sought to send to Oriental countries rather than flood our own and depreciate its fickle value. The old silver dollar by the Coinage Act of 1873 was abolished. The codification of all the legislation of Congress since the foundation of the Mint in 1792 was thus accomplished. Other needed legislative enactments were passed by Congress on his recommendations.
In 1877 Dr. Linderman wrote, and Putnam published, “Money and Legal Tender in the United States,” a valuable and interesting contribution to the science of finance, which was favorably received abroad as well as here. The same year his official report presented one of the most exhaustive, profound, and able efforts which has ever emanated from the Government press. The fact that several of his reports were in use as text books of technical information in some of the technical schools (notably that at Harvard University), will serve to show the estimation in which the late Dr. Linderman was held as an authority upon coinage, mining, and finance. When the Japanese established their mint, that government made him the liberal offer of $50,000 to stay in their country one year and organize their mint service.
When M. Henri Cernuschi, the eminent financier and the Director of the French Mint, was in this country in 1878, he said, “Dr.
Linderman’s name is as celebrated on the continent of Europe in connection with his opinions on the double standard of metallic currency, as that of Garibaldi in connection with the Italian revolution.”
In 1877 Dr. Linderman was appointed a commissioner, with power to name two others, to investigate abuses in the San Francisco Mint and Custom House. He appointed ex-Governor Low, of California, and Mr. Henry Dodge, and this commission sat as a court of inquiry in San Francisco in 1877. He returned to Washington in the autumn of that year. His report of the commission was duly approved, and all the changes it advised were made by the Government authorities.
In 1853 Dr. Linderman married Miss Emily Davis, a highly accomplished and talented lady, daughter of George H. Davis, one of the pioneer coal operators of the Wyoming and Carbon districts. Dr. Linderman died at his residence in Washington in January, 1879, after a long illness superinduced by his self-sacrificing care and solicitude for public interests. His conscientious and valuable aid and advice in counsel, his conception of public duty, which so entirely guided his conduct in all his official relations connected with our present monetary system, established through his efforts, justly entitle him to be held in grateful remembrance for the benefits he conferred upon his fellow countrymen.[20]
COL. A. LOUDON SNOWDEN, S
S ,
was born in Cumberland County, Pennsylvania, and descends from one of the old families of Pennsylvania.
He was educated at the Jefferson College in Washington, Pennsylvania. On the completion of his collegiate course he studied law, but on May 7, 1857, just before being admitted to the bar, accepted the position of Register, tendered him by his uncle, the late Hon. James Ross Snowden, then Director of the United States Mint.
