both sides of (\[eqn 7.04.1\]) are continuous, and therefore the equality above holds for all $t\leq T$ (a.s.).

\[remark 1\] Let $\sigma\in {\mathbb{R}}$. Suppose $g_n \to g$ in ${\mathbb{H}}^{\sigma}_2(T,\ell_2)$ as $n\to \infty$, and $\phi\in C^{\infty}_0 ({\mathbb{R}}^d)$. Then $(g_n (t, \cdot),\phi)_{L_2} \to (g(t, \cdot),\phi)_{L_2}$ in $L_2(\Omega\times [0,T],{\mathcal{P}}, \ell_2)$, and therefore Lemma \[lemma 2\](ii) holds with $h_n(t):=(g_n(t, \cdot),\phi)_{L_2}$ and $h(t):=(g(t, \cdot),\phi)_{L_2}$.

\[derivative\] Let $\alpha>1/2$ and $g\in {\mathbb{H}}^{\infty}_0(T,\ell_2)$. Then $I^{\alpha} \sum_{k=1}^\infty\int^{\cdot}_0 g^k (s) dW^k_s$ is differentiable in $t$ and (a.s.) for all $t\leq T$ $$\frac{\partial}{ \partial t}(I^{\alpha} \sum_{k=1}^\infty \int^{\cdot}_0 g^k (s) dW^k_s)(t)=\frac{1}{\Gamma(\alpha)}\sum_{k=1}^\infty \int^t_0 (t-s)^{\alpha-1}g^k(s)dW^k_s.$$

We integrate the right hand side and then use the stochastic Fubini’s theorem (see e.g. [@Kr11]) to get $$\begin{aligned}
\frac{1}{\Gamma(\alpha)} \sum_{k=1}^\infty\int^t_0\int^s_0 (s-r)^{\alpha-1}g^k(r)dW^k_r ds &=&\frac{1}{\Gamma(\alpha)}\sum_{k=1}^\infty \int^t_0\int^t_r (s-r)^{\alpha-1} ds g^k(r)dW^k_r\\
&=&\frac{1}{\alpha \Gamma(\alpha)} \sum_{k=1}^\infty\int^t_0 (t-r)^{\alpha}  g^k(r)dW^k_r.\end{aligned}$$ Similarly, by Lemma \[lemma 2\](i) $$\begin{aligned}
I^{\alpha} \sum_{k=1}^\infty \Big(\int^{\cdot}_0 g^k (s) dW^k_s \Big) (t)&=&\frac{1}{\Gamma(\alpha)}\sum_{k=1}^\infty\int^t_0 (t-s)^{\alpha-1}\left(\int^s_0 g^k(r)dW^k_r\right) ds\\
&=&\frac{1}{\alpha \Gamma(\alpha)} \sum_{k=1}^\infty\int^t_0 (t-r)^{\alpha}  g^k(r)dW^k_r.\end{aligned}$$ The lemma is proved.

Lemma \[derivative\] can be easily extended for any $g\in {\mathbb{L}}_2(T,\ell_2)$.

For the remainder of this paper, we assume that holds. For $a\in {\mathbb{R}}$, denote $a_+=\max\{a,0\}$. Define $$\label{gamma0}
\gamma_0:=\frac{(2\gamma-1)_+}{\beta}<2.$$ Note that since $\gamma<\beta+1/2$ we have $$\label{gamma1}
\gamma_0<2, \quad \text{and}\quad \gamma_0=0 \quad \text{if}\,\, \gamma\leq 1/2.$$

\[def 3.23\] We write $u\in {\mathcal{H}}^{\sigma+2}_2(T)$ if 