$f: V \rightarrow V$ and $\sigma: V \rightarrow L_2(V)$ are measurable and satisfy Lipschitz and linear growth conditions with constants $L_{f}$ and $L_{\sigma}$ respectively, where $L_{2}(V)$ is the family of Hilbert-Schmidt operators from $V$ to itself. $g: V \rightarrow L(V)$ and $g^{\prime} : V \rightarrow L(V, L(V))$ are Lipschitz continuous in the following senses: $$\begin{aligned}
\label{g1}
\sup _{i \in \mathbb{N}}\left|g\left(v_{1}\right) e_{i}-g\left(v_{2}\right) e_{i}\right| \leq L_{g}\left|v_{1}-v_{2}\right|,\end{aligned}$$ $$\begin{aligned}
\label{g2}
    \sup _{i \in \mathbb{N}}\left|g^{\prime}\left(v_{1}\right) e_{i}-g^{\prime}\left(v_{2}\right) e_{i}\right|_{L(V)} \leq L_{g}^{\prime}\left|v_{1}-v_{2}\right|,\end{aligned}$$ where $\left\{e_{i}\right\}_{i \in \mathbb{N}}$ is the complete orthonormal basis in $V$.

[ For $v_1, v_2, u_1, u_2\in V$, there exist $c_1,c_2>0$, such that $$\begin{aligned}
\label{growth}
\sup _{i \in \mathbb{N}}|g(v_{1}) e_{i}| &\leq& \sup _{i \in \mathbb{N}}|g(v_{1}) e_{i}-g(0) e_{i}| +\sup _{i \in \mathbb{N}}|g(0) e_{i}|\cr
&\leq&  c_1 (1+|v_{1}|),\end{aligned}$$ here $c_1:=\max\{L_{g},\sup _{i \in \mathbb{N}}|g(0) e_{i}|\}$ and by [@Rascanu2002 Lemma 7.1], $$\begin{aligned}
\label{linear}
\sup _{i \in \mathbb{N}}|g(v_{1}) e_{i}-g(v_{2})e_{i}-g(u_1) e_{i}+g(u_2) e_{i}| &\leq& c_2 |v_{1}-v_{2}-u_1+u_2|\cr
&&+c_2 |v_1-v_2|(|v_1-u_1|+|v_2-u_2|),\end{aligned}$$ holds.]{}

Taking a parameter $ 1-H<\alpha<\frac{1}{2}$, for the measurable functions $h:[0,T] \rightarrow V$, let $$\begin{aligned}
\label{alphanorm}
\|h(t)\|_{\alpha}=|h(t)|+\int_{0}^{t} \frac{|h(t)-h(s)|}{(t-s)^{\alpha+1}} \mathrm{d}s.\end{aligned}$$

Denote by $\mathcal{B}^{\alpha,2}(0, T ; V)$ the space of measurable functions $h:[0, T] \rightarrow V$ endowed with the norm $\|\cdot\|_{\alpha,T}$ defined by $$\|h\|_{\alpha,T}^2:=\sup_{t\in [0,T]}|h(t)|^2+\int_{0}^{T}\Big(\int_{0}^{t} \frac{|h(t)-h(s)|}{(t-s)^{\alpha+1}} \mathrm{d} s \Big)^2\mathrm{d}t<\infty.$$

For $\alpha \in (1-H,\frac{1}{2})$, $V$-valued process $(u_t)_{ t\in[0,T]}$, is 