We study the logical and computational strength of weak compactness in the
separable Hilbert space \ell_2.
Let weak-BW be the statement the every bounded sequence in \ell_2 has a weak
cluster point. It is known that weak-BW is equivalent to ACA_0 over RCA_0 and
thus that it is equivalent to (nested uses of) the usual Bolzano-Weierstra{\ss}
principle BW. We show that weak-BW is instance-wise equivalent to the
\Pi^0_2-CA. This means that for each \Pi^0_2 sentence A(n) there is a sequence
(x_i) in \ell_2, such that one can define the comprehension functions for A(n)
recursively in a cluster point of (x_i). As consequence we obtain that the
Turing degrees d > 0" are exactly those degrees that contain a weak cluster
point of any computable, bounded sequence in \ell_2. Since a cluster point of
any sequence in the unit interval [0,1] can be computed in a degree low over
0', this show also that instances of weak-BW are strictly stronger than
instances of BW.
We also comment on the strength of weak-BW in the context of abstract Hilbert
spaces in the sense of Kohlenbach and show that his construction of a solution
for the functional interpretation of weak compactness is optimal