Let $T>0$. Let $V$ and $H$ be real Hilbert spaces such that $V$ is separable and densely and continuously embedded in $H$. Identify $H$ with a subspace of $V^*$ by the map $h\mapsto (v\mapsto (h,v)_H)$. Let $(X_m)_{m\in\mathbb N}$ be an increasing sequence of finite-dimensional subspaces of $V$ such that $\bigcup_{m=1}^\infty X_m$ is dense in $V$. Let $B:V\times V\to\mathbb R$ be a bounded, symmetric, and coercive [bilinear form](/page/Bilinear%20Form). Thus there are constants $M>0$ and $\alpha>0$ such that \begin{align*} |B[v,z]|\le M\|v\|_V\|z\|_V \end{align*} and \begin{align*} B[v,v]\ge \alpha\|v\|_V^2 \end{align*} for all $v,z\in V$. Define $A:V\to V^*$ by \begin{align*} (Av)(z):=B[v,z]\qquad\text{for }v,z\in V. \end{align*} Let $a\ge 0$, let $f\in L^2(0,T;H)$, let $u_0\in V$, and let $u_1\in H$. Then there exists a function $u$ such that \begin{align*} u\in L^\infty(0,T;V),\qquad u_t\in L^\infty(0,T;H),\qquad u_{tt}\in L^2(0,T;V^*) \end{align*} and \begin{align*} u_{tt}+a u_t+Au=f \end{align*} in $L^2(0,T;V^*)$. Equivalently, for every $v\in V$ and every $\eta\in C_c^\infty(0,T)$, \begin{align*} -\int_0^{\mathsf T} (u_t(t),v)_H\eta'(t)\,d\mathcal L^1(t)+a\int_0^{\mathsf T} (u_t(t),v)_H\eta(t)\,d\mathcal L^1(t)+\int_0^{\mathsf T} B[u(t),v]\eta(t)\,d\mathcal L^1(t)=\int_0^{\mathsf T} (f(t),v)_H\eta(t)\,d\mathcal L^1(t). \end{align*} Moreover $u\in C_w([0,T];V)$, $u_t\in C_w([0,T];H)$, and the initial data are attained in the sense that \begin{align*} u(0)=u_0\quad\text{weakly in }V,\qquad u_t(0)=u_1\quad\text{weakly in }H. \end{align*}