Appendix A. Banach Spaces In the appendices A, B and C we summarise the fundamental concepts and results we require from the theory of Banach spaces. For the proofs, we shall refer to well-known textbooks, and we give a limited selection at the end. We assume that the reader is familiar with the fundamental topological concepts (topology, neighborhood, basis of neighborhoods, open, contact point, etc.) and give full proofs only for a few results from the theory of Banach spaces which are of central significance for our applications. As a rule, we base our considerations on a real vector space. In order to be able to deal with topological concepts such as open sets, limit points, convergent sequences, Cauchy sequences, etc. in a vector space X, such as in the Euclidean space IRn , we will assume that a generalised "concept of length" or a "seminorm" p is given in X. A s eminorm is a function p : X -+ [0, 00 ) with the properties p(>.x) = 1>'lp(x) and p(x + y) :s:; p(x) + p(y) for all >. E IR and all x, y EX. If p( x) = 0 always implies x = 0, a seminorm is called a norm, which is written as p(x) = Ilxli. A vector space X together with a norm 11·11 defined for its elements is called a normed space. In a normed space (X, 11·11, we obtain a topology, and hence all topological concepts, by defining the following sets as open sets. A subset A c X is called open if, for every x E A, there exists an r = r x > 0 such that the open ball Br(x) = {y E X Illy - xII < r} with centre x and radius r belongs to A : Br(x) C A. In a normed space (X, II . II) a sequence (Xn)nElN converges to a point x E X if, and only if, for every t: > 0 there exists an n., such that IIx n - xII < t: holds for all n ~ n.,. A sequence (Xn)nElN is called a Cauchy sequence if, and only if, for every t: > 0 there exists an n., such that for all n, m ~ ne , IIx n -xmll < t: holds. A normed space in which every Cauchy sequence converges is said to be complete, or a Banach space. A Banach space is said to be separable if there exists a countable dense subset of it. Two norms 11·111 and 11·112 on a vector space X are said to be equivalent if there exist positive numbers c, c' such that IIxlh :s:; cllxll2 and IIxl12 :s:; c'lIxlll hold for all x EX. Equivalent norms generate the same topology. A subset M of a normed space (X, 11·11) is said to be bounded if the norm is bounded on M: 364 Appendix A. Banach Spaces sup{lIxlll x EM} < 00. Equivalent norms have identical bounded sets. Com­ plete normed spaces, which can be regarded as (infinite-dimensional) analo­ gous of the Euclidean spaces lRn, ~an be obtained in the following manner: let there be a scalar product (.,.) defined on a vector space X. This is a function (. , .) : X x X -+ lR with the following properties: (x, AlYl + A2Y2) = Al (x, Yl) + A2(x, Y2), (x, y) = (y, x) for all x, Yl, Y2, Y E X and all Aj E lR, (x, x) 2: 0, and (x, x) = 0 for all x EX only when x = O. A vector space X with a scalar product (. ,.) is called a pre-Hilbert space. In a pre-Hilbert space (X, (.,.)) one has a norm defined by x -+ IIxll = +~, called the norm induced by the scalar product. If the normed space (X, 11·11 = ..;r.1) is complete, then (X, (., .)) is called a Hilbert space. The Theorem of Jordan and von Neumann characterises those norms which are induced by a scalar product. The characteristic of these norms is the parallelogram law for all Xj EX. The Cauchy-Schwarz inequality l(x,y)1 :5l1xll'lIylI for all x,y E X holds in every pre-Hilbert space. The Projection Theorem is valid in every Hilbert space just as in any finite­ dimensional Euclidean space: let M be a subspace (which means a closed vector subspace) of a Hilbert space 1i. Then every vector x E 1i can be uniquely represented as x = U x + Vx where U x E M and Vx E M..L = {x E 1i I (x, y) = 0, Vy EM}, and one has I\vxl\ = c(x,M) = inf I\x - ull = Ilx - uxll· uEM We then write 1i = M ffi M..L and say that 1i is the direct orthogonal sum of M and M..L. If the orthogonal subspace M..L of M is finite dimensional, then we say that M has finite codimension. We now come to a few examples. Example A.I. The Euclidean Spaces lRn , n E IN. One can introduce the following norms in the vector space lRn : for an arbitrary number p E [1,00) and x E lR n one sets IIxli oo = sup IXil. l~i~n It follows from the inequalities IIxll oo :5 I\xl\p :5 nl/Pllxl\oo, x E lRn, that all these norms are equivalent. (lRn, 1\. lip) is a Banach space for 1 :5 p ~ +00. It is only for p = 2 that (lRn, 1\·lIp) is a Hilbert space with the scalar product Appendix A. Banach Spaces 365 n (x,y) = LXiYi. i=l Example A.2. The Sequence Spaces [P(JR), I ::s; p ::s; 00. Let lP(JR) denote the vector space of all sequences x = (Xn)nEIN of real numbers Xn for which I ::s; p < 00 and sup IXnl < 00 for p = +00 nEIN hold. Together with the norms II x li p = (~ IxnlP) liP, l::S;p<oo, Ilxll oo = sup Ixnl , nEIN respectively, the sequences spaces (lP(JR), II· lip) are complete normed spaces. In this case different values of p give rise to nonequivalent norms. Once again, only the sequence space z2(JR) is a Hilbert space. Its scalar product is (x, y) = 2::::='=1 xnYn' For I ::s; P ::s; +00 the dual exponent p' corresponding to p is defined by the equation lip' + lip = 1. For all x E IP(JR) and all Y E [P' (JR), one has x . Y = (XnYn)nElN E 11(JR) and 00 Ilx . ylh = L IXnYnl ::s; IlxllpllYllp' (Holder's Inequality). n=l Example A.3. Spaces of Continuous Functions. Let K denote a compact topo­ logical space (see also Appendix C) and C(K; JR) = C(K), be the set of all continuous functions f : K -t JR. One knows that C(K) is a real vector space in a natural way and IlfllK = sUPxEK If(x)1 defines a norm on C(K). C(K) is complete with respect to this norm, and is thus a Banach space but not a Hilbert space. Example A.4. Spaces of Differentiable Functions. For a compact interval f = [a, b], let Cl(f; IRn) be the space of all n-tuples of real functions continuously differentiable in (a, b), which together with their derivatives possess continuous extensions to the boundary points t = a and t = b. The real vector space Cl(f, IRn) is a Banach space with respect to the norm Ilqlh,l = sup{lq(t)l, 14(t)l} . tEl For q(t) = (ql (t), ... , qn(t)), qi E Cl(f, IR), let q(t) = (ql(t), ... ,qn(t)), qi(t) = ~i(t) 366 Appendix A. Banach Spaces and Iq(t)1 = (~lqi(t)12) 1/2 A norm equivalent to II . Ih.r is given by the following: Example A.5. The Lebesgue Spaces LP(lRn). For every real number p 2 1 let CP (lR n) = {J : lRn ---+ lR I I measurable, II I summable}. £P (lRn) is a real vector space and I ---+ 1I/11p where II/l1p = (in I/(x)IP dX) lip is a seminorm on CP(lRn). One has II/l1p = 0 {:} I = 0 almost everywhere. Denoting by N the subspace of all functions from CP(lRn) which vanish almost everywhere, the quotient space together with the norm induced by 1I·llp (which is again denoted by II· lip) is a Banach space (Theorem 01 Riesz-Fischer). Similarly one defines for p = +00, COO (lRn) = {J : lRn ---+ lR I I measurable and bounded almost everywhere} 11/1100 = inf{c Ie> 0, and apart from a null set I/(x)1 ~ c holds} and, further, Then Loo(lRn) is also a Banach space. Just as in the case of sequence spaces, Holder's inequality holds here also, which means that if I E LP(lRn), 9 E U ' (lRn) with p' being the dual exponent to p 21, then I· 9 E Ll(lRn), and one has Once again we obtain a Hilbert space only for p = 2. L2(lRn) is a Hilbert space with the scalar product (f,gh = { I(x)g(x)dx. lrn. n The norm thereby induced is 11·112. (In our notation we do not make explicit the distinction between the equivalence classes and the representatives from the equivalence classes.) Appendix A. Banach Spaces 367 Replacing the space IR n by a measurable set G ~ IR n one obtains the Banach spaces LP(G), 1 ::; p ::; +00, of Lebesgue. The Banach spaces LP(G) with 1 ::; p < 00 are separable. For real vector spaces Xl and X 2 , let L(XI' X 2 ) be the collection of all linear maps or operators from X I to X 2.