Chapter 4 Appendix A: Bases in Banach spaces 4.1 Schauder bases In this section we recall some of the notions and results presented in the course on Functional Analysis in Fall 2012 [Schl]. Like every vector space a Banach space X admits an algebraic or Hamel basis,i.e.asubsetB X, so that every ⊃ x X is in a unique way the (finite) linear combination of elements in B.This 2 definition does not take into account that we can take infinite sums in Banach spaces and that we might want to represent elements x X as converging series. 2 Hamel bases are also not very useful for Banach spaces, since (see Exercise 1), the coordinate functionals might not be continuous. Definition 4.1.1. [Schauder bases of Banach Spaces] Let X be an infinite dimensional Banach space. A sequence (e ) X is n ⇢ called Schauder basis of X, or simply a basis of X, if for every x X,thereisa 2 unique sequence of scalars (an) K so that ⇢ 1 x = anen. n=1 X Examples 4.1.2. For n N let 2 N en =(0,...0 , 1, 0,...) K 2 n 1times − Then (e ) is a basis of ` ,1 p<| {z }and c . We call (e ) the unit vector of ` n p 1 0 n p and c0,respectively. Remarks. Assume that X is a Banach space and (en) a basis of X.Then 33 34 CHAPTER 4. APPENDIX A: BASES IN BANACH SPACES a) (en) is linear independent. b) span(en : n N)isdenseinX, in particular X is separable. 2 c) Every element x is uniquely determined by the sequence (an) so that x = N j1=1 anen. So we can identify X with a space of sequences in K . PropositionP 4.1.3. Let (en) be the Schauder basis of a Banach space X.For n N and x X define e⇤ (x) K to be the unique element in K,sothat 2 2 n 2 1 x = en⇤ (x)en. n=1 X Then e⇤ : X K is linear. n ! For n N let 2 n P : X span(e : j n),x e⇤ (x)e . n ! j 7! n n Xj=1 Then P : X X are linear projections onto span(e : j n) and the following n ! j properties hold: a) dim(Pn(X)) = n, b) Pn Pm = Pm Pn = Pmin(m,n),form, n N, ◦ ◦ 2 c) limn Pn(x)=x, for every x X. !1 2 Pn, n N,arecalledtheCanonical Projections for (en) and (e⇤ ) the Coordinate 2 n Functionals for (en) or biorthogonals for (en). Theorem 4.1.4. Let X be a Banach space with a basis (en) and let (en⇤ ) be the corresponding coordinate functionals and (Pn) the canonical projections. Then Pn is bounded for every n N and 2 b =sup Pn L(X,X) < , n || k 1 2N and thus e X and n⇤ 2 ⇤ Pn Pn 1 2b e⇤ = k − − k . k nkX⇤ e e k nk k nk We call b the basis constant of (ej).Ifb =1we say that (ei) is a monotone basis. Furthermore n + 1 1 : X R0 , aiei aiei =sup aiei , ||| · ||| ! 7! n j=1 j=1 2N j=1 X X X is an equivalent norm under which (ei) is a monotone basis. 4.1. SCHAUDER BASES 35 Definition 4.1.5. [Basic Sequences] Let X be a Banach space. A sequence (x ) X 0 is called a basic sequence n ⇢ \{ } if it is a basis for span(xn : n N). 2 If (ej) and (fj) are two basic sequences (in possibly two di↵erent Banach spaces X and Y ). We say that (ej) and (fj) are isomorphically equivalent if the map n n T : span(ej : j N) span(fj : j N), ajej ajfj, 2 ! 2 7! Xj=1 Xj=1 extends to an isomorphism between the Banach spaces between span(ej : j N) 2 and span(fj : j N). 2 Note that this is equivalent with saying that there are constants 0 <c C n so that for any n N and any sequence of scalars (λj) it follows that 2 j=1 n n n c λ e λ f C λ e . j j j j j j j=1 j=1 j=1 X X X Proposition 4.1.6. Let X be Banach space and (xn : n N) X 0 . Then 2 ⇢ \{ } (xn) is a basic sequence if and only if there is a constant K 1,sothatforall n ≥ m<nand all scalars (aj) K we have j=1 ⇢ m n (4.1) a x K a x . i i i i i=1 i=1 X X In that case the basis constant is the smallest of all K 1 so that (4.1) holds. ≥ Theorem 4.1.7. [The small Perturbation Lemma] Let (xn) be a basic sequence in a Banach space X, and let (xn⇤ ) be the coor- dinate functionals (they are elements of span(xj : j N)⇤)andassumethat(yn) 2 is a sequence in X such that 1 (4.2) c = x y x⇤ < 1. k n − nk·k nk n=1 X Then a) (yn) is also basic in X and isomorphically equivalent to (xn), more precisely 1 1 1 (1 c) a x a y (1 + c) a x , − n n n n n n n=1 n=1 n=1 X X X for all in X converging series x = n anxn. 2N b) If span(xj : j N) is complementedP in X, then so is span(yj : j N). 2 2 36 CHAPTER 4. APPENDIX A: BASES IN BANACH SPACES c) If (xn) is a Schauder basis of all of X, then (yn) is also a Schauder basis of X and it follows for the coordinate functionals (y ) of (y ),thaty n⇤ n n⇤ 2 span(x⇤ : j N),forn N. j 2 2 Now we recall the notion of unconditional basis. First the following Proposi- tion. Proposition 4.1.8. For a sequence (xn) in Banach space X the following state- ments are equivalent. a) For any reordering (also called permutation) σ of N (i.e. σ : N N is ! bijective) the series n xσ(n) converges. 2N b) For any ">0 thereP is an n N so that whenever M N is finite with 2 ⇢ min(M) >n, then n M xn <". 2 k P c) For any subsequence (nj) the series j xnj converges. 2N d) For sequence (" ) 1 the series P 1 " x converges. j ⇢{± } j=1 j nj In the case that above conditions hold weP say that the series xn converges unconditionally. P Definition 4.1.9. A basis (ej) for a Banach space X is called unconditional,if for every x X the expansion x = e ,x e converges unconditionally, where 2 h j⇤ i j (e ) are coordinate functionals of (ej). j⇤ P Asequence(x ) X is called an unconditional basic sequence if (x ) is an n ⇢ n unconditional basis of span(xj : j N). 2 Proposition 4.1.10. For a sequence of non zero elements (xj) in a Banach space X the following are equivalent. a) (xj) is an unconditional basic sequence, b) There is a constant C,sothatforallfiniteB N,allscalars(aj)j B K, ⇢ 2 ⇢ and A B ⇢ (4.3) a x C a x . j j j j j A j B X2 X2 c) There is a constant C0, so that for all finite sets B N,allscalars ⇢ (aj)j B K,andall("j)j B 1 ,ifK = R,and("j)j B z 2 ⇢ 2 ⇢{± } 2 ⇢{ 2 C : z =1 ,ifK = C, | | } n (4.4) " a x C0 a x . j j j j j j B j=1 X2 X 4.2. MARKUSHEVICH BASES 37 In this case we call the smallest constant C = Cs which satisfies (4.3) for all n n, A 1, 2 ...,n and all scalars (aj) K the supression-unconditional ⇢{ } j=1 ⇢ constant of (xn) and we call the smallest constant C0 = Cu so that (4.4) holds n n for all n, ("j)j=1 1 ,or("j)j=1 z C : z =1,andallscalars n ⇢{± } ⇢{ 2 | | } (aj) K the unconditional constant of (xn). j=1 ⇢ Moreover, it follows that (4.5) C C 2C . s u s Proposition 4.1.11. If (xn) be an unconditional basic sequence. Then 1 1 (4.6) C =sup a b x : x = a x B and b 1 . u i i i i i 2 X | i| n j=1 i=1 o X X Remark. While for Schauder bases it is in general important how we order them, the ordering is not relevant for unconditional bases. We can therefore index unconditional bases by any countable set. 4.2 Markushevich bases Not every separable Banach space has a Schauder basis [En]. But it has at least a bounded and norming Markushevich basis according to a result of Ovsepian and Pe lczy´nski [OP]. We want to present this result in this section, Definition 4.2.1. A countable family (en,en⇤ )n X X⇤ is called 2N ⇢ ⇥ biorthogonal, if e⇤ (em)=δ(m,n), for all m, n N, • n 2 fundamental, or complete, if span(en : i N)isdenseinX, • 2 total, if for any x X with e⇤ (x) = 0, for all n N, it follows that x = 0, • 2 n 2 norming, if for some constant c>0 it follows that • sup x⇤(x) c x , for all x X. x span(e :n ) B | |≥ k k 2 ⇤2 n⇤ 2N \ X⇤ and in that case we also say that (en,en⇤ )n is c-norming, 2N shrinking, if span(e⇤ : n N)=X⇤, and • n 2 bounded, or uniformly minimal if C =supn en en⇤ < , and we say in • 2N k k·k k 1 that case that (en,en⇤ )n is C-bounded and call C the bound of (en,en⇤ )n .