MAT 550, REAL ANALYSIS SUNY at Stony Brook, SPRING 99 Problem Set 4, REVISED VERSION Due on Friday, Febuary 26

Two Banach spaces X and Y are said to be isomorphic if there is a linear map L : X → Y which is 1-1, onto and preserves norms, i.e., ∥Lx∥_{Y} = ∥x∥_{X} for every x ∈ X. As usual, let ℓ^{p} denote the L^{p} space on the positive integers with respect to counting measure. The following exercises combine to show that none of the spaces ℓ^{p}, 1 ≤ p < ∞ is isomorphic to any of the others (in fact, none is isomorphic to a closed subspace of any of the others). If you can’t do one of the exercises, you may assume it for the subsequent ones.

A sequence {x_{n}} in a Banach space X is called a Schauder basis if for every x ∈ X there is a unique sequence of scalars {a_{n}} such that x = ∑_{n} a_{n}x_{n}. For example, let e_{n} ∈ ℓ^{p} be the sequence which is all 0’s except for a 1 in the nth position. Then {e_{n}} is a Schauder basis for each of the spaces ℓ^{p}, 1 ≤ p < ∞. A sequence {x_{n}} is called a basic sequence if it is a Schauder basis for its closed linear span. A sequence is called normalized if ∥x_{n}∥_{X} = 1 for all n. Two basic sequences, {x_{n}} of X and {y_{n}} of Y are called equivalent if the series ∑_{n=1}^{∞} a_{n}x_{n} converges iff ∑_{n=1}^{∞} a_{n}y_{n} converges. Suppose {x_{n}} is a Schauder basis of X. A block basis of {x_{n}} is a sequence of non-zero vectors of the form u_{k} = ∑_{n=p_{k}+1}^{p_{k+1}} a_{n}x_{n} with {a_{n}} scalars and {p_{k}} an increasing sequence of integers. A sequence {y_{n}} ⊂ L^{p}(μ) is said to converge weakly to zero if ⟨y_{n}, x⟩ = ∫ y_{n}xdμ → 0 for every x in the dual space L^{q}, \frac{1}{p} + \frac{1}{q} = 1 (equivalently, L(y_{n}) → 0 for every bounded linear functional L on L^{p}).

1. Suppose {x_{n}} is a normalized Schauder basis in X. Suppose {y_{n}} is a basic sequence in X with ∑_{n=1}^{∞} ∥x_{n} − y_{n}∥_{X} < ∞. Show that {x_{n}} and {y_{n}} are equivalent. (You may assume X = ℓ^{p} and the sum is strictly less than 1 if you wish.)

2. Show that the usual basis {e_{n}} in ℓ^{p} converges weakly to zero. Show that if T : ℓ^{p_{1}} → Y ⊂ ℓ^{p_{2}} is an isomorphism then the image sequence {y_{n}} = {T (e_{n})} converges weakly to zero in ℓ^{p_{2}}. Deduce the following: suppose that with respect to the standard basis {e_{n}} we have the expansions y_{n} = ∑_{k=1}^{∞} a_{k,n}e_{k}. Show that for any N > 0, lim_{n→∞} ∥ ∑_{k=1}^{N} a_{k,n}e_{k}∥_{p_{2}} = 0.

3. Suppose {y_{n}} ⊂ ℓ^{p} is as in the previous problem. Show there is a normalized block basis {u_{k}} of {e_{n}} which is equivalent to some subsequence of the {y_{n}}.

4. Suppose 1 ≤ p < ∞. Suppose {u_{n}} is a normalized block basis of the standard basis {e_{n}}. Then {e_{n}} is equivalent to {u_{n}}.

5. If ℓ^{p} is isomorphic to any closed subspace Y of ℓ^{q}, prove the standard bases {e_{n}} in ℓ^{p} and ℓ^{q} are equivalent. Show this is impossible unless p = q (and thus ℓ^{p} cannot be isomorphic to a closed subsapce of ℓ^{q} unless p = q).