By N. L. Carothers

It is a brief direction on Banach area concept with designated emphasis on yes facets of the classical idea. particularly, the path makes a speciality of 3 significant issues: The easy thought of Schauder bases, an creation to Lp areas, and an creation to C(K) areas. whereas those themes might be traced again to Banach himself, our basic curiosity is within the postwar renaissance of Banach house conception caused by means of James, Lindenstrauss, Mazur, Namioka, Pelczynski, and others. Their based and insightful effects are priceless in lots of modern learn endeavors and deserve higher exposure. when it comes to necessities, the reader will want an basic realizing of sensible research and at the very least a passing familiarity with summary degree conception. An introductory path in topology may even be priceless, even though, the textual content encompasses a short appendix at the topology wanted for the path.

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**Extra resources for A Short Course on Banach Space Theory**

**Example text**

An ∈ R. 8. Given T ∈ B(X, Y ), show that ker T = ⊥ (range T ∗ ), the annihilator of range T ∗ in X , and that ker T ∗ = (range T )⊥ , the annihilator of range T in Y ∗ . 9. Given T ∈ B(X, Y ), show that T ∗∗ is an extension of T to X ∗∗ in the → Y ∗∗ denote the canonical following sense: If i : X − → X ∗∗ and j : Y − ∗∗ embeddings, prove that T (i(x)) = j(T (x)). In short, T ∗∗ (xˆ ) = T x. 10. Let S be a dense linear subspace of a Banach space X , and let T : S − →Y be a continuous linear map, where Y is also a Banach space.

If X is a Banach space with X = M ⊕ N and if we write Q : X − →X for the projection with kernel M, then our earlier observations show that X/(ker Q) = X/M is isomorphic to N . Conversely, if the quotient map q : X− → X/M is an isomorphism on some closed subspace N of X , then Q = (q| N )−1 q, considered as a map from X to X , deﬁnes a projection with range N. In the special case of linear functionals, these observations tell us that (X/M)∗ = M ⊥ (the annihilator of M in X ∗ ). That is, the dual of X/M can be → identiﬁed with the functionals in X ∗ that vanish on M.

27. Let M be a closed subspace of a Banach space X and let q : X − → X/M ∗ denote the quotient map. Prove that q is an isometry from (X/M)∗ into X ∗ with range M ⊥ . Thus (X/M)∗ can be identiﬁed with M ⊥ . 28. Let M be a closed subspace of a Banach space X and let i : M − →X denote the inclusion map. Prove that i ∗ is a quotient map from X ∗ onto M ∗ with kernel M ⊥ . Conclude that M ∗ can be identiﬁed with X ∗ /M ⊥ . 29. Let M be a closed subspace of a normed space X . For any f ∈ X ∗ , show that min{ f − g : g ∈ M ⊥ } = sup{| f (x)| : x ∈ M, x ≤ 1}.