By Belton A.

Those notes are an extended model of a suite written for a path given to final-yearundergraduates on the college of Oxford.A thorough realizing of the Oxford third-year b4 research path (an introductionto Banach and Hilbert areas) or its identical is a prerequisite for this fabric. Weuse [24] as a compendium of effects from that sequence of lectures. (Numbers in squarebrackets check with goods within the bibliography.)The writer recognizes his debt to all these from whom he has learnt functionalanalysis, specially Professor D. A. Edwards, Dr G. R. Allen and Dr J. M. Lindsay. Thestudents attending the path have been very necessary, specially Mr A. Evseev, Mr L. Taitzand Ms P. Iley.This rfile was once typeset utilizing LATEX2ε with Peter Wilson’s memoir classification and theAMS-LATEX and XY-pic programs. The index was once produced via the MakeIndexprogram.This variation encompasses a few extra workouts and the digital model is equippedwith links, due to the hyperref package deal of Sebastian Rahtz and Heiko Oberdiek.

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By Zorn’s lemma we conclude that there exists (h, P ) ∈ S that is maximal for ; if we can show that P = X then we are done. Suppose otherwise; then P is a proper subspace of X and there exists a proper extension of h, by the first part of this proof. This contradicts the maximality of (h, P ). 9. (Bohnenblust-Sobczyk) Let p be a seminorm on the vector space X and suppose that M is a subspace of X. , |Φ(x)| p(x) for all x ∈ X). 34 Dual Spaces Proof Suppose first that X is a real vector space. Note that a seminorm is a sublinear functional, so we may apply the Hahn-Banach theorem to obtain Φ ∈ X ′ such that Φ|M = φ and Φ(x) p(x) for all x ∈ X, but also −Φ(x) p(−x) = p(x) ∀ x ∈ X by the homogeneity of the seminorm p.

Let M be a finite-dimensional subspace of the normed space X and let N be a closed subspace of X such that X = M ⊕ N. Prove that if φ0 is a linear functional on M then φ : M ⊕ N → F; m + n → φ0 (m) ∀ m ∈ M, n ∈ N is an element of the dual space X ∗ . 3. Prove that a normed vector space X is separable if its dual X ∗ is. ] Find a separable Banach space E such that E ∗ is not separable. ] Prove that a reflexive Banach space E is separable if and only if E ∗ is. 4. reflexive. 5. Prove that any infinite-dimensional normed space has a discontinuous linear functional defined on it.

12. Let L1 (R) denote the space of (equivalence classes of) complex-valued, Lebesgue-integrable functions on the real line, with norm · 1: L1 (R) → R+ ; f → R |f |. This is a commutative Banach algebra when equipped with the convolution product: f ⋆ g : R → R; t → R f (t − s)g(s) ds. ) This algebra lacks a unit; it is easy u to see that L1 (R) is isomorphic to the algebra given by adjoining the Dirac measure δ0 : by definition (f ⋆ δ0 )(t) = “ R f (t − s)δ0 (s) ds ” = f (t) ∀t ∈ R and δ0 ⋆ δ0 = δ0 .