Weak convergence in vector-valued Hilbert space Let $V$ be a separable Hilbert space and define $X=L^2(0,T;V)$. Then $u_m\to u$ weakly in $X$ means

for every $v\in X'=L^2(0,T;V')$
  $$
\int_0^T\langle v(t),u_m(t)\rangle\  dt\to\int_0^T\langle v(t),u(t)\rangle\ dt\tag{1}
$$
  where $\langle\rangle$ is the dual pair $(V',V)$.

If one assumes that

for every $v\in V'$
  $$
\int_0^T\langle v,u_m(t)\rangle\  dt\to\int_0^T\langle v,u(t)\rangle\ dt\tag{2}
$$

then it is not generally true that $u_m\to u$ weakly. The case $V=\mathbb{R}$ gives a counterexample. 
Would (2) together with $\{u_m\}$ being bounded in $X$ be enough to imply (1)? (I don't have a way to test it.)
[Added: Are there any "handy" extra assumptions that would allow (2) imply (1)?]
 A: It seems to me that there is a trivial counterexample: just take $u_m(t)=u(t)$ where 
$$
\int_0^T u(t)\, dt =0, $$ 
and $u(t)$ is not identically $0$. This is a bounded sequence which does not converge weakly to $0$, however 
$$
\int_0^T \langle v, u_m(t)\rangle dt = 0\qquad \forall v\in V'.$$
A: A subset $A\subset X'$ is called total if it is not contained in any proper closed subspace.
Note that $A$ is total in $X'$ iff its anihilator in $X$ is trivial.
Your space $X$ is a separable Hilbert space, thus closed bounded sets are compact and metrizable for the weak topology.
In particular, every bounded sequence has a weakly converging subsequence.
The following could be therefore proven easily by a sub-sub-sequence argument:
Assume $\{u_m\}$ is bounded in $X$ and $A$ total in $X'$. Then $u_m\to u$ weakly is equivalent to
$(*)$ $\langle u_m,v\rangle \to \langle u,v\rangle$ for every $v\in A$.
Note that if $A$ is not total in $X'$ then the condition $(*)$ does not guarantee weak convergence (for example, take $u_m=0$ for all $m$ and $u$ a nontrivial vector in the annihilator of $A$).
Now, back to our specific question. The set of constant functions is a proper closed subspace of $X'$, hence not total. Therefore the answer is "no".
What you are really seem to be asking is: what are good examples of total subsets of $X'$ that I can use? 
Given a total subset $B\subset V$, each of the sets
$$ \{ \chi_{[a,b]}\cdot v\mid 0<a<b<1,~ v\in B\} $$
and 
$$ \{ f\cdot v\mid f\in C([0,1)),~v\in B\} $$ is total in $X'$ and might be suitable for your purposes. The proof is an exercise.
