By N. L. Carothers

ISBN-10: 0521842832

ISBN-13: 9780521842839

It is a brief path on Banach house thought with detailed emphasis on convinced features of the classical thought. specifically, the path specializes in 3 significant issues: The straight forward idea of Schauder bases, an creation to Lp areas, and an creation to C(K) areas. whereas those subject matters could be traced again to Banach himself, our fundamental curiosity is within the postwar renaissance of Banach area concept led to through James, Lindenstrauss, Mazur, Namioka, Pelczynski, and others. Their based and insightful effects are worthwhile in lots of modern study endeavors and deserve higher exposure. when it comes to necessities, the reader will desire an hassle-free figuring out of useful research and at the very least a passing familiarity with summary degree thought. An introductory direction in topology may even be precious, even though, the textual content encompasses a short appendix at the topology wanted for the direction.

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Before we can describe the method, we’ll need a few preliminary facts. Given two Banach spaces X and Y , we can envision their sum X ⊕ Y as the space of all pairs (x, y), where x ∈ X and y ∈ Y . Up to isomorphism, it doesn’t much matter what norm we take on X ⊕ Y . ” This is a simple consequence of the fact that all norms on R2 are equivalent. ) Given a sequence of Banach spaces X 1 , X 2 , . . , we deﬁne the p -sum of X 1 , X 2 , . . to be the space of all sequences (xn ), with xn ∈ X n , for which p (xn ) p = ∞ n=1 x n X n < ∞, in case p < ∞, or (x n ) ∞ = supn x n X n < ∞, in case p = ∞, and we use the shorthand (X 1 ⊕ X 2 ⊕ · · ·) p to denote this new space.

How is this possible? Well, i=1 of ﬁnite codimension and so must contain a nonzero vector. The claim is that any such norm one x will do. To see this, choose y ∈ S F , any scalar λ ∈ R, 34 Disjointly Supported Sequences in L p and 35 p and estimate y + λx ≥ yi + λx − y − yi ≥ yi + λx − ε/2, for some i ≥ yi∗ (yi + λx) − ε/2 1 . = 1 − ε/2 ≥ 1+ε Thus, y ≤ (1 + ε) y + λx , for all λ, whenever y = 1. Since the inequality is homogeneous (λ being arbitrary), this is enough. 2. Every inﬁnite-dimensional Banach space contains a closed subspace with a basis.

Since the dyadic rationals are dense in [0, 1], it’s not hard to see that the f n have dense linear span. Thus, ( f n ) is a viable candidate for a basis for C[0, 1]. If we set pn = nk=0 ak f k , then pn ∞ = max | pn (tk )| 0≤k≤n because pn is a polygonal function with nodes at t0 , . . , tn . And if we set pm = m k=0 ak f k for m > n, then we have pm (tk ) = pn (tk ) for k ≤ n because f j (tk ) = 0 for j > n ≥ k. Hence, pn ∞ ≤ pm ∞ . This implies that ( f n ) is a normalized basis for C[0, 1] with basis constant K = 1.

### A Short Course on Banach Space Theory by N. L. Carothers

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