Let $(\Omega,\mathcal{F},P)$ be a probability space and let $(E,\mathcal{E})$ be a separable Hilbert space ($E$) with Borel $\sigma$-algebra $\mathcal{E}$. For concreteness let us set $E=L^{2}[a,b]$ with the usual inner product.

For a sequence of stochastic processes $\{X_{N}(t)\}_{N=1}^{\infty}$ and $X(t)$ whose sample paths belong to $L^{2}[a,b]$, suppose I have shown that

$a)$ Convergence of finite-dimensional distributions, i.e. that

$$(X_{N}(t_{1}),\ldots,X_{N}(t_{k})) \to (X(t_{1}),\ldots,X(t_{k})), \qquad N \to \infty$$

in distribution, for all $(t_{1},\ldots,t_{k}) \in [a,b]^{k}$, any $k \in \mathbb{N}$ and

$b)$ Tightness of the family of random elements $\{X_{N}(t)\}_{N=1}^{\infty}$ in $L^{2}[a,b]$, i.e. that for all $\epsilon>0$ there is a compact $K \subset L^{2}[a,b]$ such that

$$\mu_{N}(K) > 1-\epsilon$$

uniformly in $N$, where $\mu_{N}$ is the probability measure on $L^{2}[a,b]$ induced by the $X_{N}$.

$\textbf{My question:}$ Are conditions $a)$ and $b)$ enough to conclude that for all bounded continuous functionals $f : L^{2}[a,b] \to \mathbb{R}$, one has the weak convergence

$$\mathbb{E}(f(X_{N})) \to \mathbb{E}(f(X)), \qquad N \to \infty.$$

From what I can tell in the literature, this type of result is usually applied to spaces like $C[0,1]$ (continuous functions on $[0,1]$) (see e.g. Billingsley's book "convergence of probability measures"). But for more general function spaces (e.g. $L^{2}[a,b]$) I can't find a good reference. Is it still true here? Are there interesting examples of function spaces where this claim is false?