MIT OpenCourseWarehttp://ocw.mit.edu 18.102 Introduction to Functional Analysis Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.� |� � � PROBLEM SET 5 FOR 18.102, SPRING 2009 DUE 11AM TUESDAY 17 MAR. RICHARD MELROSE You should be thinking about using Lebesgue’s dominated convergence at several points below. Problem 5.1 Let f : R −→ C be an element of L1(R). Define � ( )f x x ∈ [−L, L](5.1) fL(x) = 0 otherwise. Show that fL ∈ L1(R) and that fL − f| → 0 as L → ∞. Problem 5.2 Consider a real-valued function f : R −→ R which is locally integrable in the sense that (5.2) gL(x) = f(x) x ∈ [−L, L] 0 x ∈ R \ [−L, L] is Lebesgue integrable of each L ∈ N. (1) Show that for each fixed L the function ⎧ ⎪⎨ ⎪⎩ gL(x) if gL(x) ∈ [−N, N ] N if gL(x) > N −N if gL(x) < −N (N )(5.3) gL (x) = is Lebesgue integrable. (N)(2) Show that 0 as N → ∞. (3) Show that there is a sequence, hn, of step functions such that|g − gL|L → (5.4) hn(x) f(x) a.e. in R.→ (4) Defining ⎧⎪⎪⎪⎨ ⎪⎪⎪⎩ 0 x �∈ [−L, L] hn(x) if hn(x) ∈ [−N, N ], x ∈ [−L, L](N)(5.5) hn,L = . N if hn(x) > N, x ∈ [−L, L] −N if hn(x) < −N, x ∈ [−L, L] Show that |h(N) (N) n,L − g 0 as n → ∞.| →L 1� � � � � 2 RICHARD MELROSE Problem 5.3 Show that L2(R) is a Hilbert space. First working with real functions, define L2(R) as the set of functions f : R −→ R which are locally integrable and such that f2 is integrable. | |(N) (N)(1) For such f choose hn and define gL, gL and hn by (5.2), (5.3) and (5.5). (2) Show using the sequence h(N) for fixed N and L that g(N) and (g(N))2 are � n,L L L (N) (N)in L1(R) and that |(hn,L )2 − (gL � )2| → 0 as n → ∞. (3) Show that (�gL)2 ∈ L1(R) and that |(g(N))2 − (gL)2| → 0 as N → ∞.L (4) Show that |(gL)2 − f2| → 0 as L → ∞. (5) Show that f, g ∈ L2(R) then fg ∈ L1(R) and that (5.6) | fg| ≤ |fg| ≤ �f �L2 �g�L2 , �f�L2 2 = |f|2 . (6) Use these constructions to show that L2(R) is a linear space. (7) Conclude that the quotient space L2(R) = L2(R)/N , where N is the space of null functions, is a real Hilbert space. (8) Extend the arguments to the case of complex-valued functions. Problem 5.4 Consider the sequence space ⎧ ⎫ ⎨ �⎬ (5.7) h2,1 = ⎩ c : N � j �−→ cj ∈ C; (1 + j2)|cj |2 < ∞⎭ . j (1) Show that (5.8) h2,1 × h2,1 � (c, d) �−→ �c, d� = (1 + j2)cj dj j is an Hermitian inner form which turns h2,1 into a Hilbert space. (2) Denoting the norm on this space by � · �2,1 and the norm on l2 by � · �2, show that (5.9) h2,1 ⊂ l2 , �c�2 ≤ �c�2,1 ∀ c ∈ h2,1 . Problem 5.5 In the separable case, prove Riesz Representation Theorem directly. Choose an orthonormal basis {ei} of the separable Hilbert space H. Suppose T : H −→ C is a bounded linear functional. Define a sequence (5.10) wi = T (ei), i ∈ N. (1) Now, recall that |T u| ≤ C�u�H for some constant C. Show that for every finite N, N(5.11) |wi|2 ≤ C2 . j=1� 3 PROBLEMS 5 (2) Conclude that {wi} ∈ l2 and that (5.12) w = wiei ∈ H. i (3) Show that (5.13) T (u) = �u, w�H ∀ u ∈ H and �T � = �w�H . Department of Mathematics, Massachusetts Institute of
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