TITLE: Approximation via finite rank Cameron-Martin projections
QUESTION [10 upvotes]: Let $(W, \|\cdot\|_W)$ be a real separable Banach space equipped with
a non-degenerate Gaussian Borel measure $\mu$.  Let $H \subset W$ be
the corresponding Cameron-Martin Hilbert space (also known as the
reproducing kernel Hilbert space or RKHS), which is dense in $W$; let
$\langle \cdot, \cdot \rangle_H$ denote the inner product on $H$.
Then the inclusion map $i : H \to W$ is continuous, injective and has
dense range, so the same is true of its adjoint $i^* : W^* \to H$.
Let $W_* \subset H \subset W$ be the image of $i^*$.
Suppose $\{e_j\}$ is an orthonormal basis for $H$ which is contained
in $W_*$, so that $e_j = i^* f_j$ for some $\{f_j\} \subset W^*$.
Define the finite rank projection $P_n : W \to W_*$ by $P_n x =
\sum_{j=1}^n f_j(x) e_j$.
Note that if $h \in H$ then $P_n h = \sum_{j=1}^n \langle e_j, h
\rangle e_j$.  So the restriction $P_n|_H$ is orthogonal projection
and we have $P_n h \to h$ in $H$-norm.  In other words, $P_n|_H \to
I_H$ strongly on $H$, where $I_H$ is the identity map on $H$.
Moreover, it can be shown that $P_n \to I_W$ in
$L^2(W, \mu; W)$.  In other words, $\int_W \|P_n x - x\|_{W}^2 \mu(dx)
\to 0$.  In particular, there is a subsequence $P_{n_k}$ such that for
$\mu$-almost every $x \in W$, we have $P_{n_k} x \to x$ in $W$-norm.  (This is essentially the fact that $\|\cdot\|_W$ is a measurable norm  on $H$ in the sense of Gross.  I believe this result is due to Dudley, Feldman and Le Cam.  A short proof due to Daniel Stroock can be found in these lecture notes by Bruce Driver; see Theorem 44.8.) 
Now in some cases much more is true.  For instance, consider the
standard Wiener measure on $W = C_0((0,1])$ (i.e. all continuous paths
starting at 0).  Then $H$ is the Sobolev space $H^1_0((0,1])$.  If we
take $e_j$ to be the Schauder (aka Faber-Schauder) basis (i.e. the
integrals of the Haar wavelets), then $P_n x$ is a piecewise linear
function agreeing with $f$ at its vertices, which include all the
dyadic rationals as $n \to \infty$.  So $P_n x \to x$ uniformly,
i.e. in $W$-norm, for every $x$ (not just almost every).  In other
words, $P_n \to I$ strongly on $W$.
As a related, not-quite example, let $W = \mathbb{R}^\infty$, viewed
as the space of all real sequences with the product topology, and let
$\mu$ be standard product Gaussian measure (I know this is not a Banach
space).  Then $H = \ell^2$.  If we choose the standard basis $\{e_j\}$
for $H$, then $P_n x$ is merely the sequence that agrees with $x$ in
its first $n$ terms and is 0 thereafter.  Clearly $P_n x \to x$ in the
$W$ topology, for every $x$.
I suppose this can't always happen, so I am looking for a
counterexample.

What is an example of $W$, $\mu$ and $\{e_n\}$ such that it is not
    the case that $P_n \to I$ strongly on $W$?  Better, is there an
    example where there is no subsequence $P_{n_k}$ with $P_{n_k} \to I$
    strongly on $W$?

REPLY [3 votes]: For an example, take for $W$ any separable Banach space that fails the approximation property, or even just does not have a Schauder basis.  Every $\ell_p$ for $p\not= 2$ has such a subspace; see, for example, volume 1 of Classical Banach Spaces by Lindenstrauss and Tzafriri.