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.75 Problems 6: Due 11AM Tuesday, 31 Mar Hint: Don’t pay too much attention to my hints, sometimes they are a little off-the-cuff and may not be very helpfult. An example being the old hint for Problem 6.2! Problem 6.1 Let H be a separable Hilbert space. Show that K ⊂ H is compact if and only if it is closed, bounded and has the property that any sequence in K which is weakly convergent sequence in H is (strongly) convergent. Hint:- In one direction use the result from class that any bounded sequence has a weakly convergent subsequence. Problem 6.2 Show that, in a separable Hilbert space, a weakly convergent se-quence {vn}, is (strongly) convergent if and only if the weak limit, v satisfies (12.19) �v�H = lim n→∞ �vn�H . Hint:- To show that this condition is sufficient, expand (12.20) (vn − v, vn − v) = �vn�2 − 2 Re(vn, v) + �v�2 . Problem 6.3 Show that a subset of a separable Hilbert space is compact if and only if it is closed and bounded and has the property of ‘finite dimensional approxi-mation’ meaning that for any � > 0 there exists a linear subspace DN ⊂ H of finite dimension such that (12.21) d(K, DN ) = sup inf {d(u, v)} ≤ �. u∈Kv∈DN Hint:- To prove necessity of this condition use the ‘equi-small tails’ property of compact sets with respect to an orthonormal basis. To use the finite dimensional approximation condition to show that any weakly convergent sequence in K is strongly convergent, use the convexity result from class to define the sequence {vn�}in DN where vn�is the closest point in DN to vn. Show that vn�is weakly, hence strongly, convergent and hence deduce that {vn} is Cauchy. Problem 6.4 Suppose that A : H −→ H is a bounded linear operator with the property that A(H) ⊂ H is finite dimensional. Show that if vn is weakly convergent in H then Avn is strongly convergent in H. Problem 6.5 Suppose that H1 and H2 are two different Hilbert spaces and A : H1 −→ H2 is a bounded linear operator. Show that there is a unique bounded linear operator (the adjoint) A∗ : H2 −→ H1 with the property (12.22) �Au1, u2�H2 = �u1, A∗u2�H1 ∀ u1 ∈ H1, u2 ∈
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