Let $H$ together with an inner product $\langle \cdot, \cdot \rangle$ be a real Hilbert space. Let $|\cdot|$ be the induced norm. I'm reading Theorem 7.4 (Hille-Yosida) in Brezis' Functional Analysis, i.e.,
Let $A: D(A) \subset H \to H$ be a maximal monotone (unbounded linear) operator. Then, given any $u_0 \in D(A)$ there exists a unique function $$ u \in C^1([0,+\infty) ; H) \cap C([0,+\infty) ; D(A)) $$ satisfying $$ (6) \quad \begin{cases} \frac{d u}{d t}+A u&=0 \quad \text{on} \quad [0,+\infty), \\ u(0)&=u_0. \end{cases} $$ Moreover, $$ |u(t)| \leq\left|u_0\right| \quad \text { and } \quad\left|\frac{d u}{d t}(t)\right|=|A u(t)| \leq\left|A u_0\right| \quad \forall t \geq 0. $$
In above theorem, $D(A)$ in $C([0,+\infty) ; D(A))$ has the subspace topology induced by $|\cdot|$. We define a new inner product on $D(A)$ by $$ \langle u, v \rangle_{D(A)} := \langle u, v \rangle+\langle Au, Av \rangle. $$
Then $D(A)$ together with $\langle \cdot, \cdot \rangle_{D(A)}$ is a Hilbert space.
Is it true that the solution $u$ belongs to $C \big ( [0,+\infty) ; (D(A), \langle \cdot, \cdot \rangle_{D(A)}) \big )$?
Thank you so much for your elaboration!
The norm on $D(A)$ is the graph norm $\|u\|_0 = |u| + |Au|$ or the equivalent Hilbert norm $\|u\|_1 = \sqrt{|u|^2 + |Au|^2}$ which comes from your inner product; see footnote 2 on p. 185 in Brezis' book..
Therefore the answer to your question is yes.