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Library of Congress Cataloging-in-Publication Data
Names: André, Robert (Mathematician), author.
Title: Point-set topology with topics : basic general topology for graduate studies / Robert André, University of Waterloo, Canada.
Description: New Jersey : World Scientific, [2024] | Includes bibliographical references and index.
Identifiers: LCCN 2023021913 | ISBN 9789811277337 (hardcover) | ISBN 9789811277344 (ebook for institutions) | ISBN 9789811277351 (ebook for individuals)
Subjects: LCSH: Topology--Textbooks. | Point set theory--Textbooks.
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Definition1.1 Let V beavectorspaceoverthereals.An inner product isanoperationwhichmapspairsofvectorsin V toareal number.Wedenoteaninnerproducton V by v, w .Areal-valued functionon V × V isreferredtoasan innerproduct ifandonlyifit satisfiesthefollowingfouraxioms:
IP1: Thenumber v, v isgreaterthanorequalto0forall v in V . Equalityholdsifandonlyif v =0.(Hence,if v isnot0, v, v is strictly largerthan0.)
IP2: Forall v, w ∈ V , u, v = v, u (Commutativity)
IP3: Forall u, v, w ∈ V , u + w, v = u, v + w, v
IP4: Forall u, v ∈ V ,and α ∈ R, αu, v = α u, v
Avectorspace V iscalledan innerproductspace ifitisequipped withsomespecifiedinnerproduct.
Definition1.2 Let V beaninnerproductspace.If v isavectorin V wedefine v = v, v
Theexpression v iscalledthe norm (orlength)ofthevector v inducedbytheinnerproduct u, v on V .
Example1. Itiseasilyverifiedthat,for x =(x1 ,x2 ,x3 ,...,xn )and y =(y1 ,y2 ,y3 ,...,yn )in Rn ,itswell-knowndot-product x, y = x y = x1 y1 + x2 y2 + + xn yn
x = (x1 ,x2 ,x3 ,...,xn ) = (x1 ,x2 ,x3 ,...,xn ) · (x1 ,x2 ,x3 ,...,xn ) = n i=1 x2 i
Thisparticularnormisreferredtoasthe Euclideannorm on Rn . Itisalsoreferredtoasthe L2 -normon Rn ,inwhichcase,itwill berepresentedas x 2 .Wewillusethisparticularnormtomeasure distancesbetweenvectorsin Rn .Thatis,thedistancebetween x = (x1 ,x2 ,x3 ,...,xn )and y =(y1 ,y2 ,y3 ,...,yn )isdefinedtobe x y = n i=1 (xi yi )2
Inthecasewhere n =2or3,thisrepresentstheusualdistance formulabetweenpointsin2-spaceand3-space,respectively.Inthe casewhere n =1,itrepresentstheabsolutevalueofthedifference oftwonumbers.
Example2. Considerthevectorspace, V = C [a,b],thesetofall continuousreal-valuedfunctionsontheclosedinterval[a,b]equipped withtheusualadditionandscalarmultiplicationoffunctions.We definethefollowinginnerproducton C [a,b]as:
= b a
(x)g (x) dx
ShowingthatthisoperationsatisfiestheinnerproductaxiomsIP1 toIP4isleftasanexercise.Inthiscase,thenormof f ,inducedby thisinnerproduct,isseentobe f = b a f (x)2 dx
Itisalsoreferredtoasthe L2 -normon C [a,b]andwerepresentitas f 2 .
Theorem1.3(Cauchy–Schwarzinequality). Let V beavector spaceequippedwithaninnerproductanditsinducednorm.Then, forvectors x and y in V, | x, y |≤ x y
Equalityholdstrueifandonlyif x and y arecollinear(i.e., x = αy or y = αx).
Proof. Thestatementclearlyholdstrueif y =0. Let x, y betwo(notnecessarilydistinct)vectorsin V where y =0. Foranyrealnumber t, 0 ≤ x ty, x ty = x 2 2t x, y + t2 y 2
Choosing t = x, y y 2 intheaboveequationweobtain 0 ≤ x 2 x, y 2 y 2
Theinequality, | x, y |≤ x y ,follows.
Wenowprovethesecondpartofthestatement.If x and y are collinear,say x = αy ,then | x, y | = | αy, y | = |α| y, y = |α| y y = αy y = x y
Conversely,suppose | x, y | = x y .If y iszerothen 0= ty and so x and y arecollinear.Suppose y = 0.Consider t = x, y 2 y 2 . x ty, x ty = x 2 x, y 2 y 2 (asdescribedabove) =0
So,byIP1, x ty, x ty =0implies x ty =0so x and y are collinear.Wehaveshownthatequalityholdsifandonlyif x and y arecollinear.
Wenowverifythatanorm, ,whichisinducedbyaninnerproduct , onthevectorspace V willalwayssatisfythefollowingthree fundamentalproperties:
(1)Forall x ∈ V , x ≥ 0,equalityholdsifandonlyif x =0.
(2)Forall x ∈ V andscalar α ∈ R, αx = |α| x .
(3)Forall x, y ∈ V , x + y ≤ x + y .
Thefirstpropertyfollowsfromthefactthatthenormisdefinedas beingthesquarerootofanumber.Thesecondpropertyfollowsfrom thestraightforwardargument: αx = αx,αx = √α2 x, x = |α| x
Corollary1.4 Thetriangleinequalityfornormsinducedbyinner products. Foranypairofvectors x and y inaninnerproductspace, x + y ≤ x + y .Ifequalityholdsthenthetwovectors x and y arecollinear.
Proof.
(Cauchy–Schwarz) =( x + y )2
Thus x + y ≤ x + y ,asrequired. Inthecasewherewehaveequality:Supposewehave x + y = x + y .Then x + y 2 =( x + y )2 .Fromthedevelopmentabove,theinequalitiesmustbeequalities,sowemusthave x, y = x y .Thismeans | x, y | = x ||y .BytheCauchy–Schwarztheorem, x and y arecollinear.
Example3. Usethethedotproductpropertiesandthelawof cosines, c2 = a2 + b2 2ab cos θ (where a,b,c representthelengths ofthesidesofatriangle ABC and θ istheanglebetweentothe sides AB and AC )toprovetheCauchy–Schwarzinequalityin R2 .
Solution: Considerthetwovectors a and b in R2 whereoneisnota scalarmultipleoftheother.Thentheextremitiesofthetwovectors formatrianglewithangle θ between a and b.Thenthenorms a , b and b a arenumberswhichcanrepresentthelengthofthesides ofthetriangle.Let θ representtheanglebetween a and b.
Weapplythecosinelawtoobtain b a 2 = a 2 + b 2 2 a b cos θ
If a, b representsthedot-product,weobtain,byapplyingthedotproductpropertiesandtheproperty a 2 = a, a
Example4. Suppose a and b =(1, 0)aretwovectorsin R2 where a =1withanglebetween a and b = θ .Showthat a, b =cos θ ,as isrepresentedinFigure1.1.
