Bases in Banach Spaces

1 SCHAUDER BASES

Let X be an infinite-dimensional Banach space over the field of real or complex numbers. When viewed as a vector space, X is known to possess a Hamel basis — a linearly independent subset of X that spans the entire space. Unfortunately, such bases cannot in general be constructed, their very existence depending on the axiom of choice, and their usefulness is therefore severely limited. Of far greater importance and applicability in analysis is the notion of a basis first introduced by Schauder [1927].

Definition. A sequence of vectors {x1, x2, x3, …} in an infinite-dimensional Banach space X is said to be a Schauder basis for X if to each vector x in the space there corresponds a unique sequence of scalars {c1, c2, c3, …} such that

=∑n=1∞cnxn.

The convergence of the series is understood to be with respect to the strong (norm) topology of X; in other words,

−∑i=1ncixi→0asn→∞

Henceforth, the term basis for an infinite-dimensional Banach space will always mean a Schauder basis.

Example. The Banach space lp (1 p < ∞) consists, by definition, of all infinite sequences of scalars c = {c1, c2, c3, …} such that p=∑n=1∞|cn|p1/p<∞. The vector operations are coordinatewise. In each of these spaces, the “natural basis” {e1, e2, e3, …}, where

n=00....010…,

and the 1 appears in the nth position, is easily seen to be a Schauder basis. If c = {cn} is in lp, then the obvious expansion =∑n=1∞cnen is valid.

It is clear that a Banach space with a basis must be separable. Reason: If {xn} is a basis for X, then the set of all finite linear combinations cnxn, where the cn are rational scalars, is countable and dense in X. It follows, for example, that since l∞ is not separable, it cannot possess a basis.

The “basis problem”— whether or not every separable Banach space has a basis — was raised by Banach [1932] and remained until recently one of the outstanding unsolved problems of functional analysis. The question was finally settled by Per Enflo [1973], who constructed an example of a separable Banach space having no basis. The negative answer to the basis problem is perhaps surprising in light of the fact that bases are now known for almost all the familiar examples of infinite-dimensional separable Banach spaces.

PROBLEMS

1. Prove that every vector space has a Hamel basis.

2. Prove that every Hamel basis for a given vector space has the same number of elements. This number is called the (linear) dimension of the space.

3. Show that a Hamel basis for an infinite-dimensional Banach space is uncountable.

4. Show that the dimension of l∞ is equal to c. (Hint: Show that the set {(1, r, r2, …): 0 < r < 1} is linearly independent.)

5. Let X be an infinite-dimensional Banach space.

a. Prove that dim X c. (Hint: Show that there is a vector space isomorphism between l∞ and a subspace of X.)

b. Prove that if X is separable, then dim X = c.

6. The Banach space c0 consists of all infinite sequences of scalars which converge to zero (with the l∞ norm). Show that the natural basis is a Schauder basis for c0.

7. Exhibit a Schauder basis for the Banach space c consisting of all convergent sequences of scalars (with the l∞ norm).

8. Aninfinite series ∑ xn in a Banach space X is said to be unconditionally convergent if every arrangement of its terms converges to the same element. It is said to be absolutely convergent if the series ‖xn‖ is convergent. Show that every absolutely convergent series in X is unconditionally convergent. What about the converse?

9. A basis {xn} for a Banach space X is said to be unconditional (absolute) if every convergent series of the form cnxn is unconditionally (absolutely) convergent.

(a) Show that the natural basis is unconditional for the spaces lp, 1 p < ∞, and c0. Show also that it is absolute for lp only when p = 1. Is it absolute for c0?

(b) Show that the sequence of vectors

,0,0,0…,1100…,1110…,…

forms a basis for c0 which is not unconditional.

2 SCHAUDER’S BASIS FOR C[a, b]

One of the most important and widely studied classical Banach spaces is C[a, b], the space of all continuous functions on the closed finite interval [a, b], together with the norm

f‖=max|fx|.

The celebrated Weierstrass approximation theorem asserts that the polynomials are dense in C[a, b]: if f is continuous on [a, b], then for every positive number ε there is a polynomial P such that the inequality

fx−Px|<ε

holds throughout the interval [a, b].

For a given continuous function, a sequence of approximating polynomials can even be given explicitly. The most elegant representation is due to Bernstein. Let us suppose, for simplicity, that f is continuous on the interval [0, 1]. Then the nth Bernstein polynomial for f is

nx=∑k=0nnkfknxk1−xn−k,n=1,2,3,....

As is well known,

x=limn→∞Bnx

uniformly on [0, 1] (see Akhiezer [1956, p. 30]).

Since every polynomial can be uniformly approximated on a closed interval by a polynomial with rational coefficients, the preceding remarks show that the space C[a, b] is separable; in fact, it has a basis.

Proof. We are going to construct a basis for C[a, b] consisting of piecewise-linear functions fn (n = 0, 1, 2, …). This means that to each function f in the space there will correspond a unique sequence of scalars {cn} such that

x=∑n=0∞cnfnx

uniformly on [a, b].

Let {x0, x1, x2, …} be a countable dense subset of [a, b] with x0 = a and x1 = b. Set

0x=1andf1x=x−ab−a.

When n 2, the set of points {x0, x1, …, xn–1} partitions [a, b] into disjoint open intervals, one of which contains xn; call it I. Define

nx=0ifx∉I1ifx=xnlinearelsewhere

for n = 2, 3, 4, … The sequence {f0, f1, f2, …} will be the required basis.

For each function f in C[a, b] and each positive integer n, we denote by Ln f the polygonal function that agrees with f at each of the points x0, x1,…, xn; we denote by L0 f the function whose constant value is f(x0). Since f is uniformly continuous on [a, b], a simple continuity argument shows that

nf→funiformlyonab.

Therefore, we can write

=L0f+∑n=1∞Lnf−Ln−1f.

We are going to show that there are scalars c1, c2, c3, … such that

nf−Ln−1f=cnfnn=1,2,3,….

For this purpose, we shall define a sequence of functions {g0,...