The Problems for the Second Test for 18.102 Brief Solutions

Let DN be the linear space spanned by the ej for j ≤ N then if the orthogonal decomposition H = DN ⊕D⊥ N gives u = u′ + u′′ and the distance d(u,DN ) = ‖u′′‖ so the finite dimensional approximation property follows from (2). Conversely, suppose that K is closed and bounded and has the FDAP. By the assumed boundedness, any sequence in K has a weakly convergent subsequence. Denote such a sequence un. For each DN let PN be the orthogonal projection onto DN . The sequence PNun is weakly convergent in DN (because bounded maps preserve weak convergence) but since this is finite dimensional, PNun is convergent, and hence Cauchy. If DN satisfies (1) for /3, and n is large enough then for m ≥ n,