Vtyjkyuk

I wonder if the following is true

$$ L_mathrmloc^infty (mathbbR_+ ; H^1 (mathbbR^3)) cap C(mathbbR_+ ; H^-1 (mathbbR^3)) subset C_mathrmw (mathbbR_+ ; H^1 (mathbbR^3)). hspace2cm (1) $$

For me $psi in C_mathrmw (mathbbR_+ ; H^1 (mathbbR^3))$ means $forall L in H^-1 (mathbbR^3)$,

$$ left| langle psi (t_n) - psi (t) , L rangle_H^1 , H^-1 right| xrightarrown rightarrow infty 0 ~~~~ mathrmwhen ~~~~ t_n xrightarrown rightarrow infty t. $$

I can think I can conclude (1) via the following little argument.

For $L in H^-1$, let $v_L$ denote the $H^1$-function given by the Riesz Representation theorem for $H^1$, i.e.,

$$ L (u) = langle u , L rangle_H^1 , H^-1 = langle u , v_L rangle_H^1. $$

Similarly, for $v in H^1$ let $L_v$ denote the functional determined by $v$ via the Riesz Representation theorem. Then, since $psi in C (mathbbR_+ ; H^-1 )$, we have

$$ left| langle psi (t_n) - psi (t) , L rangle_H^1 , H^-1 right| = left| langle v_L , L_psi (t_n) - L_psi (t) rangle_H^1 , H^-1 right| leq | L_psi (t_n) - L_psi (t) |_H^-1 | v_L |_H^1 rightarrow 0. $$

Is my reasoning correct, or am I misunderstanding the spaces and convergences involved?

Let $u$ be given as $uin L^infty(I, X)$ and $uin C(bar I,Y)$ for an intervall $I$, reflexive Banach space $X$ and normed space $Y$ with continuous embedding $Xhookrightarrow Y$. Then $uin C_w(bar I,X)$.

Take $t_nto t$. Then $(u(t_n))$ is bounded in $X$, hence contains a weakly converging subsequence $u(t_n_k)$ converging weakly to some $tilde u$ in $X$. Due to the continuity in $Y$, we have $tilde u= u(t)$. This means the weak subsequential limit does not depend on the subsequence. Moreover, each subsequence of $(u(t_n))$ contains another subsequence converging to $u(t)$ weakly in $X$. This implies $u(t_n)rightharpoonup u(t)$ in $X$, which is the weak continuity.