Chapter 3. Normed vector spaces - Trinity College Dublin

1 Chapter 3. Normed vector spacesLecture notes for MA2223P. / 15 The definition of a normDefinition NormSupposeXis a vector space over the fieldF=RorF=C. A normonXis a real-valued function||x||with the following vector :||x||= 0if and only ifx= factors:|| x||=| | ||x||for all Fand allx inequality:||x+y|| ||x||+||y||for allx,y Normed vector space (X,|| ||)consists of a vector spaceXand anorm||x||. One generally thinks of||x||as the length is easy to check that every norm satisfies||x|| 0for allx Normed vector space (X,|| ||)is also a metric space (X, d), asone may define a metricdusing the formulad(x,y) =||x y||.

2 Thisparticular metric is said to be induced by the / 15 Examples of Normed vector spacesGiven anyp 1, we can define a norm onRkby letting||x||p=[k i=1|xi|p]1 spaceC[a, b]has a similar norm for anyp 1, namely||f||p=[ ba|f(x)|pdx]1 is also a norm onRkfor the casep= . It is defined by||x|| = max1 i k|xi|.Finally, there is a similar norm onC[a, b]which is given by||f|| = maxa x b|f(x)|.3 / 15 Sequence spaces pThe space pconsists of all real sequencesx={xn}such that n=1|xn|p< .It is a Normed vector space for anyp 1and its norm is given by||x||p=[ n=1|xn|p]1 space consists of all bounded real sequencesx={xn}.

3 It isa Normed vector space and its norm is given by||x|| = supn 1|xn|.The spacec0consists of all real sequences{xn}which converge is easily seen to be a subspace of .4 / 15 Continuity of operationsTheorem Product normSupposeX, Yare Normed vector spaces . Then one may define a normon the productX Yby letting||(x,y)||=||x||+||y||.Theorem Continuity of operationsThe following functions are continuous in any Normed normf(x) =||x||, wherex vector additiong(x,y) =x+y, wherex,y scalar multiplicationh( ,x) = x, where Fandx shall mainly use this theorem to justify computations such aslimn ||xn||= limn xn.

4 5 / 15 Bounded linear operatorsDefinition Bounded, linear, continuousLetX, Ybe Normed vector spaces over the fieldF=RorF= functionT:X Yis called a linear operator, ifT(x+y) =T(x) +T(y),T( x) = T(x)for allx,y Xand all scalars functionT:X Yis called bounded, if there exists a realnumberM >0such that||T(x)|| M||x||for allx functionT:X Yis called continuous, if it is continuouswith respect to the metrics which are induced by the linear operator is also known as a linear definition, every linear operatorTis such thatT(0) = / 15 Bounded means continuousTheorem Bounded means continuousSupposeX, Yare Normed vector spaces and letT:X Ybe continuous if and only ifTis Norm of an operatorSupposeX, Yare Normed vector spaces .

5 Then the setL(X, Y)of allbounded, linear operatorsT:X Yis itself a Normed vector fact, one may define a norm onL(X, Y)by letting||T||= supx6=0||T(x)||||x||.It is easy to check that||T(x)|| ||T|| ||x||for allx also has||S T|| ||S|| ||T||wheneverS, T L(X, X).7 / 15 Norm of an operator: Example 1 Consider the right shift operatorR: p pwhich is defined byR(x1, x2, x3, ..) = (0, x1, x2, ..).This operator is easily seen to be linear and we also have||R(x)||p=||x||pfor allx particular, the norm of this operator is equal to||R||= left shift operatorL: p pis similarly defined byL(x1, x2, x3.)

6 = (x2, x3, x4, ..).Since||L(x)||p ||x||pfor allx p, we find that||L|| 1. On theother hand, we also have||L(x)||p=||x||pwheneverx1= 0and thisimplies that||L|| 1. We may thus conclude that||L||= / 15 Norm of an operator: Example 2 Suppose thatT: (Rn,|| ||1) (Rm,|| || )is left multiplication bythem nmatrixA. We then have||T(x)|| = maxi jaijxj maxi,j|aij| j|xj|= maxi,j|aij| ||x||1and this implies that||T|| maxi,j|aij|.On the other hand, the standard unit vectorx=ejsatisfies||T(x)|| ||x||1= maxi jaijxj = maxi|aij|,so we also have||T|| maxi|aij|for eachj.

7 We conclude that||T||= maxi,j|aij|.9 / 15 Finite-dimensional vector spacesSuppose thatXis a vector space with basisx1,x2, .. ,xk. Thenevery elementx Xcan be expressed as a linear combinationx=c1x1+c2x2+..+ckxkfor some uniquely determined coefficientsc1, c2, .. , ck Euclidean normSuppose thatXis a vector space with basisx1,x2, .. ,xk. Then onemay define a norm onXusing the formulax=k i=1cixi= ||x||2= k i=1|ci| norm is also known as the Euclidean or standard norm / 15 Equivalent normsDefinition Equivalent normsWe say that two norms|| ||aand|| ||bof a Normed vector spaceXare equivalent, if there exist constantsC1, C2>0such thatC1||x||a ||x||b C2||x||afor allx Equivalence of all normsThe norms of a finite-dimensional vector spaceXare all norms|| ||1and|| || are not equivalent inC[a, b]

8 Because thisspace is complete with respect to only one of the two fact,|| ||pand|| ||qare not equivalent inC[0,1]whenp < q. Toprove this, one may definefn(x) =xnfor eachn Nand then checkthat the quotient||fn||q/||fn||pis unbounded asn .11 / 15 Banach spacesDefinition Banach spaceA Banach space is a Normed vector space which is also completewithrespect to the metric induced by its Examples of Banach spaces1 Every finite-dimensional vector spaceXis a Banach sequence space pis a Banach space for any1 p .3 The spacec0is a Banach space with respect to the|| || a Banach space , thenL(X, Y)is a Banach spaceC[a, b]is a Banach space with respect to the|| || is not complete with respect to the|| ||pnorm when1 p <.

9 Suppose thatXis a Banach space and letYbe a subspace itself a Banach space if and only ifYis closed / 15 Convergence of seriesDefinition Convergence of seriesSuppose that{xn}is a sequence in a Normed vector spaceX. We saythat the series n=1xnconverges, if the partial sumsN= Nn=1xnconverges asN . If that is the case, then we denote its limit bylimN sN= limN N n=1xn= n= say that n=1xnconverges absolutely, if n=1||xn|| Absolute convergence implies convergenceSuppose thatXis a Banach space and let n=1xnbe a series whichconverges absolutely inX. Then this series must also / 15 Invertible linear operatorsDefinition InvertibilityA bounded linear operatorT:X Xis called invertible, if there is abounded linear operatorS:X Xsuch thatS T=T S=Iisthe identity operator onX.

10 If such an operatorSexists, then we callit the inverse ofTand we denote it byT Geometric seriesSuppose thatT:X Xis a bounded linear operator on a BanachspaceX. If||T||<1, thenI Tis invertible with inverse n= Set of invertible operatorsSupposeXis a Banach space . Then the set of all invertible boundedlinear operatorsT:X Xis an open subset ofL(X, X).14 / 15 Dual spaceDefinition Dual spaceSupposeXis a Normed vector space overR. Its dualX is then theset of all bounded linear operatorsT:X R, namelyX =L(X,R).Theorem Dual ofRkThere is a bijective mapT:Rk (Rk) that sends each vectoratothe bounded linear operatorTadefined byTa(x) = ki= Dual of pSuppose1< p < and letq=p/(p 1).