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