Functional analysis lecture notes by Ward T.B.

By Ward T.B.

So one can succeed in the extra attention-grabbing and precious principles, we will undertake a reasonably brutal method of a few early fabric. long proofs will occasionally be skipped over, although complete types could be made to be had. by means of the top of the path, you'll have an exceptional realizing of normed vector areas, Hilbert and Banach areas, xed element theorems and examples of functionality areas. those principles should be illustrated with purposes to di erential equations.

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1) It is clear that T ≥ 0 since it is defined as the supremum of a set of non– negative numbers. If T = 0 then T x Y = 0 for all x, so T x = 0 for all x – that is, T = 0. (2) The triangle inequality is also clear: T + S = sup (T + S)x ≤ sup x =1 T x + sup x =1 Sx = T + S . x =1 (3) λT = sup x =1 (λT )x = |λ| sup x =1 T x = |λ| T . Finally, assume that Y is a Banach space and let (Tn ) be a Cauchy sequence in B(X, Y ). Then the sequence is bounded: there is a constant K with Tn x ≤ K x for all x ∈ X and n ≥ 1.

This means that (iv) implies (i). 11. Every Hilbert space has an orthonormal basis. Any orthonormal basis in a separable Hilbert space is countable. 2. Classical Fourier analysis comes about using the orthonormal basis {e2πint }n∈Z for L2 [0, 1]. Proof. Let H be a Hilbert space, and consider the classes of orthonormal sets in H with the partial order of inclusion. 1 there exists a maximal orthonormal set K. Since K is maximal, it is complete and is therefore an orthonormal basis. 5. GRAM–SCHMIDT ORTHONORMALIZATION 57 Now let H be separable, and suppose that {xα } is an uncountable orthonormal basis.

Hence L1 [a, b] ⊃ L2 [a, b] ⊃ · · · ⊃ L∞ [a, b]. In the theorem we allow p and q to be anything in [1, ∞] with the obvious 1 interpretation of ∞ . 4[3], where we saw that 1 ⊂ 2 ⊂ · · · ⊂ ∞. 46 4. INTEGRATION Two easy consequences of H¨older’s inequality are the Cauchy–Schwartz inequality, fg 1 ≤ f 2 f +g p ≤ f p g 2 and Minkowski’s inequality, + g p. The most useful general result about Lebesgue integration is Lebesgue’s dominated convergence theorem. 3. e. e. Then f dµ = lim E n→∞ fn dµ. 2. [1] Prove that the Lp –norm is strictly convex for 1 < p < ∞ but is not strictly convex if p = 1 or ∞.

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