Let $(\xi_n)_{n\ge 1}$ be the polygonal partial sums processes built
on the linear processes $X_n=\sum_{i\ge 0}a_i(\epsilon_{n-i})$,
n ≥ 1, where $(\epsilon_i)_{i\in\mathbb{Z}}$ are
i.i.d., centered random elements in some
separable Hilbert space $\mathbb{H}$ and the ai's are bounded linear
operators $\mathbb{H}\to \mathbb{H}$, with $\sum_{i\ge 0}\lVert a_i\rVert<\infty$. We
investigate functional central limit theorem for $\xi_n$ in the
Hölder spaces $\mathrm{H}^o_\rho(\mathbb{H})$ of functions
$x:[0,1]\to\mathbb{H}$ such that ||x(t + h) - x(t)|| = o(p(h))
uniformly in t, where p(h) = hαL(1/h), 0 ≤ h ≤ 1
with 0 ≤ α ≤ 1/2 and L slowly varying at infinity. We
obtain the $\mathrm{H}^o_\rho(\mathbb{H})$ weak convergence of $\xi_n$ to
some $\mathbb{H}$ valued Brownian motion under the optimal assumption that
for any c>0, $tP(\lVert \epsilon_0\rVert>ct^{1/2}\rho(1/t))=o(1)$ when
t tends to infinity, subject to some mild restriction on L in
the boundary case α = 1/2. Our result holds in particular with
the weight functions p(h) = h1/2lnβ(1/h), β > 1/2>.