Evolution triples and the parabolic energy space

Let $V$ and $H$ be real Hilbert spaces and let $i:V\to H$ be an injective, continuous linear map with dense range. Let $V^{\ast }$ be the continuous real linear dual of $V$. Identify $H$ with its Hilbert dual by its inner product and embed $H$ into $V^{\ast }$ by $h⟼(v⟼(h,iv{)}_H)\mathrm{.}$ The resulting dense continuous embeddings are written $V\hookrightarrow H\cong H^{\ast }\hookrightarrow V^{\ast }$ and $(V,H,V^{\ast })$ is called an evolution triple. The pairing between $V^{\ast }$ and $V$ is denoted $⟨\cdot ,\cdot {⟩}_{V^{\ast },V}$.

For a Hilbert space $X$, $L^2(0,T;X)$ is the space of equivalence classes of strongly measurable $X$-valued functions $w$ with ${\int }_0^T\mathrm{\parallel }w(t){\mathrm{\parallel }}_X^{2}dt<\mathrm{\infty }\mathrm{.}$ Strong measurability means being an almost-everywhere pointwise limit of measurable simple functions. For $u\in L^2(0,T;V)$ and $g\in L^2(0,T;V^{\ast })$, say that $g$ is the weak time derivative of $u$ when ${\int }_0^T(iu(t),iv{)}_H{\varphi }^{\mathrm{\prime }}(t)dt=-{\int }_0^T⟨g(t),v{⟩}_{V^{\ast },V}\varphi (t)dt$ for every $v\in V$ and every $\varphi \in C_c^{\mathrm{\infty }}(0,T)$. The derivative is unique up to equality almost everywhere. The parabolic energy space is $W(0,T;V):=\{u\in L^2(0,T;V):u^{\mathrm{\prime }}\in L^2(0,T;V^{\ast })\},$ with norm $\mathrm{\parallel }u{\mathrm{\parallel }}_{W(0,T;V)}:=(\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2(0,T;V)}^2+\mathrm{\parallel }{u}^{\mathrm{\prime }}{\mathrm{\parallel }}_{L^2(0,T;V^{\ast })}^2{)}^{1\mathrm{/}2}\mathrm{.}$ No endpoint values are part of this definition; they are assigned only after a continuous $H$ trace representative has been proved. Separability is an additional hypothesis when it is needed by a later result.