Chapter 3 Bases in Banach Spaces 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) ⇢ 2 linear combination of elements in B. This 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 (with possibly infinite 2 non zero elements). Hamel bases are also not very useful for Banach spaces, since (see Exercise 1 the coordinate functionals might not be continuous. 3.1 Schauder Bases Definition 3.1.1. (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,there 2 is a unique sequence of scalars (an) K so that ⇢ 1 x = anen. n=1 X Examples 3.1.2. 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 n p 1 0 n basis of `p and c0,respectively. Remarks. Assume that X is a Banach space and (en) a basis of X. 65 66 CHAPTER 3. 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 = j1=1 anen. So we can identify X with a space of sequences in N, for which a e converges in X. K P n n Proposition 3.1.3.PLet X be a normed linear space and assume that (en) ⇢ X has the property that each x X can be uniquely represented as a series 2 1 x = anen, with (an) K ⇢ n=1 X (we could call (en) Schauder basis of X but we want to reserve this term only if X is a Banach space). For n N and x X define e⇤ (x) K to be the unique element in K, 2 2 n 2 so that 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 n ! j following 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 Conversely if (Pn : n N) is a sequence of linear projections satisfying 2 (a), (b), and (c), and moreover are bounded, and if e P (X) 0 and 1 2 1 \{ } en Pn(X) (Pn 1),withen =0,ifn>1, then each x X can be 2 \N − 6 2 uniquely represented as a series 1 x = anen, with (an) K, ⇢ n=1 X so in particular (en) is a Schauder basis of X in case X is a Banach space. 3.1. SCHAUDER BASES 67 Proof. The linearity of en⇤ follows from the unique representation of every x X as x = 1 e (x)e , which implies that for x and y in X and 2 j=1 n⇤ n ↵,β K, 2 P n n ↵x + βy =lim↵ e⇤(x)ej + β e⇤(y)ej n j j !1 Xj=1 Xj=1 n 1 =lim (↵e⇤(x)+βe⇤(y))ej = (↵e⇤(x)+βe⇤(y))ej, n j j j j !1 Xj=1 Xj=1 and, on the other hand 1 ↵x + βy = ej⇤(↵x + βy)ej, Xj=1 thus, by uniqueness, e⇤(↵x + βy)=↵e⇤(x)+βe⇤(y), for all j N.The j j j 2 conditions (a), (b) and (c) are clear. Conversely, assume that (Pn) is a sequence of bounded and linear projec- tions satisfying (a), (b), and (c). By (b) Pn 1(X)=Pn Pn 1(X) Pn(X), − ◦ − ⇢ for n N (put P0 = 0)and, thus, by (a), the codimension of Pn 1(X)inside 2 − Pn(X) is 1. So if e1 P1(X) 0 and en Pn(X) (Pn 1), if n>1, 2 \{ } 2 \N − then for x X,by(b) 2 Pn 1(Pn(x) Pn 1(x)) = Pn 1(x) Pn 1(x)=0, − − − − − − and thus Pn(x) Pn 1(x) (Pn 1) and − − 2N − Pn(x) Pn 1(x)=Pn(Pn(x) Pn 1(x)) Pn(X), − − − − 2 and therefore Pn(x) Pn 1(x) Pn(X) (Pn 1). Thus, we can write − − 2 \N − Pn(x) Pn 1(x)=anen, for n N, and it follows from (c) that (letting − − 2 P0 = 0) n n 1 x =limPn(x)= lim Pj(x) Pj 1(x)= lim ajej = ajej. n n − n !1 !1 − !1 Xj=1 Xj=1 Xj=1 In order to show uniqueness of this representation of x assume x = j1=1 bjej. From the continuity of Pm Pm 1, for m N it follows that − − 2 P n amem =(Pm Pm 1)(x)= lim(Pm Pm 1) bjej = bmem, − n − − !1 − ⇣ Xj=1 ⌘ and thus am = bm. 68 CHAPTER 3. BASES IN BANACH SPACES Definition 3.1.4. (Canonical Projections and Coordinate functionals) Let X be a normed linear space and assume that (ei) satisfies the assump- tions of Proposition 3.1.3. The linear functionals (en⇤ ) as defined in Proposi- tion 3.1.3 are called the Coordinate Functionals for (en) and the projections Pn, n N, are called the Canonical Projections for (en). 2 Proposition 3.1.5. Suppose X is a normed linear space and assume that (e ) X has the property that each x X can be uniquely represented as a n ⇢ 2 series 1 x = anen, with (an) K. ⇢ n=1 X If the canonical projections are bounded, and, moreover, supn N Pn < 2 k k 1 (i.e. uniformly the Pn are bounded), then (ei) is a Schauder basis of its completion X˜. ˜ ˜ ˜ Proof. Let Pn : X X, n N, be the (by Proposition 1.1.5 and Exercise ! 2 1 in Section 1.2 uniquely existing) extensions of Pn.SincePn has finite dimensional range it follows that P˜ (X˜)=P (X) = span(e : j n)is n n j finite dimensional and, thus, closed. (P˜n) satisfies therefore (a) of Proposi- tion 3.1.3. Since the Pn are continuous, and satisfy the equalities in (b) of Proposition 3.1.3 on a dense subset of X˜, (b) is satisfied on all of X˜. Finally, (c) of Proposition 3.1.3 is satisfied on a dense subset of X˜, and we deduce ˜ forx ˜ X,˜x =limk xk,withxk X, for k N, that 2 !1 2 2 x˜ P˜n(˜x) x˜ xk +sup Pj x˜ xk + xk Pn(xk) k − kk − k j N k kk − k k − k 2 and, since (Pn) is uniformly bounded, we can find for given ">0, k large enough so that the first two summands do not exceed ", and then we choose n N large enough so that the third summand is smaller than ". It follows 2 therefore that also (c) is satisfied on all of X˜. Thus, our claim follows from the second part of Proposition 3.1.3 applied to X˜. Our goal is now to show the converse of Proposition 3.1.3, and prove that if (en) is a Schauder basis, then the canonical projections are uniformly bounded, and thus that the coordinate functionals are bounded. Theorem 3.1.6. 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 N || k 1 2 3.1. SCHAUDER BASES 69 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 there is an equivalent renorming of (X, ) for which ||| · ||| k·k (e ) is a monotone basis for (X, ). n ||| · ||| Proof. For x X we define 2 x =sup Pn(x) , ||| ||| n N k k 2 since x =limn Pn(x) , it follows that x x < for x X. k k !1 k k k k||| ||| 1 2 It is clear that is a norm on X. Note that for n N ||| · ||| 2 Pn =sup Pn(x) ||| ||| x X, x 1 ||| ||| 2 ||| ||| =supsup Pm Pn(x) x X, x 1 m N k ◦ k 2 ||| ||| 2 =supsup Pmin(m,n)(x) 1. x X, x 1 m N k k 2 ||| ||| 2 Thus the projections Pn are uniformly bounded on (X, ). Let X˜ ˜ ||| · ||| be the completion of X with respect to , Pn, for n N,the(unique) ||| · ||| 2 extension of Pn to an operator on X˜. We note that the P˜n also satisfy the conditions (a), (b) and (c) of Proposition 3.1.3. Indeed (a) and (b) are purely algebraic properties which are satisfied by the first part of Proposition 3.1.3. Moreover for x X then 2 (3.1) x Pn(x) =sup Pm(x) Pmin(m,n)(x) ||| − ||| m N k − k 2 =sup Pm(x) Pn(x) 0ifn , m n k − k! !1 ≥ which verifies condition (c).