$u\in {\mathbb{H}}^{\sigma+2}_2(T)$, $u(0)\in U^{\sigma+1}_2$, and for some $f\in {\mathbb{H}}^{\sigma}_2(T)$ and $g\in {\mathbb{H}}^{\sigma+\gamma_0}_2(T,\ell_2)$ it holds that $$\label{eqn 7.08.1}
 \partial^{\beta}_tu (t,x)=f (t,x)
    +\sum_{k=1}^{\infty} \partial^{\gamma}_t \int^t_0 g^k (s,x)\, dW^k_s$$ in the sense of distributions. That is, for any $\phi\in C^{\infty}_0 ({\mathbb{R}}^d)$ the equality $$\label{sense of solution}
 (I^{1-\beta}(u-u(0))(t), \phi)_{L_2}=\int^t_0(f(s, \cdot), \phi)_{L_2}ds+\sum_{k=1}^{\infty} (I^{1-\gamma} \int^{\cdot}_0 (g^k(s, \cdot),\phi)_{L_2}\, dW^k_s)(t)$$ holds for all $t\leq T$ (a.s.). In this case we write $$f={\mathbb{D}}u, \quad g={\mathbb{S}}u$$ and define $$\label{e:3.10}
 \|u\|_{{\mathcal{H}}^{\sigma+2}_2(T)}=\|u(0)\|_{U^{\sigma+1}_2}+\|u\|_{{\mathbb{H}}^{\sigma+2}_2(T)}+\|{\mathbb{D}}u\|_{{\mathbb{H}}^{\sigma}_2(T)}+\|{\mathbb{S}}u\|_{{\mathbb{H}}^{\sigma+\gamma_0}_2(T,\ell_2)}.$$ Finally define $$\begin{aligned}
\label{e:ch0}
 {\mathcal{H}}^{\sigma+2}_{2,0}(T)={\mathcal{H}}^{\sigma+2}_2(T) \cap \{u: u(0)=0\}.
 \end{aligned}$$

By (\[e:1.10\]), (\[2014.1.23.2\]) and Lemma \[derivative\], (\[sense of solution\]) is equivalent to $$\begin{aligned}
 (u(t, \cdot)-u(0, \cdot), \phi)_{L_2}&=&\frac{1}{\Gamma(\beta)}\int^t_0(t-s)^{\beta-1}(f(s, \cdot), \phi)_{L_2}ds\\
 &&+\frac{1}{\Gamma(1+\beta-\gamma)}\sum_{k=1}^{\infty} \int^t_0 (t-s)^{\beta-\gamma}  (g^k(s, \cdot),\phi)_{L_2}\, dW^k_s.
 \end{aligned}$$

\[Banach\] [(i)]{} ${\mathcal{H}}^{\sigma+2}_2(T)$ is a Banach space.

[(ii)]{} Let $u\in {\mathcal{H}}^{\sigma+2}_2(T)$. Then $u$ is a continuous $H^{\sigma}_2$-valued process.

[(iii)]{} Assume that $u\in {\mathcal{H}}^{2}_2(T)$ and (\[eqn 7.08.1\]) holds. Then (a.s.) $$\label{eqn 8.27.1}
(k_{1-\beta}*\|u-u(0)-v\|^2_{L_2})(t)\leq 2\int^t_0 (f(s, \cdot),u(s, \cdot)-u(0, \cdot)-v(s, \cdot))_{L_2}ds \quad \hbox{for }  t\leq T ,$$ where $$\begin{aligned}
\label{e:defv}
v(t,x)=\Gamma(1+\beta-\gamma)^{-1}\sum_{k=1}^\infty \int^t_0 (t-s)^{\beta-\gamma} g^k(s,x) dW^k_s.\end{aligned}$$

\(i) We only need to prove the completeness of the space. Let $u_n$, $n=1,2,\cdots$, be a Cauchy sequence in ${\mathcal{H}}^{\sigma+2}_2(T)$ with $$f_n={\mathbb{D}}u_n, \quad g_n={\mathbb{S}}u_n.$$ Then there exist $u, f, g, u_0$ so that $u_n, f_n, g_n, u_n(0)$ converge to $u,f,g,u_0$ respectively in their 