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PROBLEM SET 8, 18.155 DUE MIDNIGHT FRIDAY 11 NOVEMBER, 2011 (1) Show that in a separable Hilbert space every bounded sequence has a weakly convergent subsequence. (2) Show that a sequence un in a separable Hilbert space H such that hun , φi converges in C for each φ ∈ H is weakly convergent. (3) A sequence of bounded operators An ∈ B(H) on a Hilbert space is said to converge strongly if An u converges in H for each u ∈ H. Show that the limits of these sequences define a bounded operator A ∈ B(H). (4) Write out a proof of the Open Mapping Theorem – that a continuous surjective linear map between Banach spaces is open, meaning that the image of an open set is open. You may use Baire’s theorem. (5) Define a closed (possibly) unbounded operator on a Hilbert space H to be a linear map A : D −→ H where D ⊂ H is a dense subspace with graph Γ = {(u, Au) ∈ H × H; u ∈ D} a closed subspace. Show (problaby using the global regularity result for elliptic operators that you might otherwise overlook, that if P (D) is elliptic of order m and u ∈ H N (Rn ) is such that P (D)u ∈ H s (Rn ) for some s ≥ N − m then u ∈ H s+m (Rn )) that n X 2 n 2 n (1) ∆ : H (R ) −→ L (R ), ∆ = Dj2 i=1 is a closed unbounded operator. 1