So, this is really late, the semester took its toll on me, and again nothing works.
The first problem is an application of the Eberlain-Smulyan theorem; let be a countable dense subset of
and note that for each
there exists a subsequence
such that
is weakly convergent. We choose the subsequences so that
and form a diagonal sequence from these and call it
.Let
and
and let
. Choose
large enough so that


For the second problem:
We first take a subsequence which we do not re-label so that where
. We then use Mazur’s lemma to write down a convex combination of the
, call them
that converge strongly to
for every
. This is important that we create these sequences truncating off the first
elements, since then this will allow us to pass the limit in the inequalities we will later obtain. More precisely, there exists
with
such that
converges strongly to
and so for every
,


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