"Double" Density of Bochner Space

Question

Let us consider the Bochner space $L^2((0,T),V)$ where $V$ is some separable Banach space and a family of Banach spaces $\{V_h\}_{h>0}$ satisfying the density property \begin{equation*} \overline{\bigcup_{h>0} V_h}^{\|\cdot\|_V} = V. \end{equation*} This could be the case for finite element approximations for some Sobolev space. It is well-known that the space of simple (piecewise constant) $V$-valued functions is dense in $L^2((0,T),V)$. Is there a similar results with $V_h$-valued simple functions? Or at least a density property of the space $L^2((0,T),V_h)$ in $L^2((0,T),V)$?

Answer 1

The simple $V_h$-valued functions are dense in the simple $V$-valued functions under the $L^2((0,T),V)$ metric. To see this, if $f = \sum_{k=1}^K v_k \chi_{A_k}$ is a simple $V$-valued function, let $V_h \ni v_{k,n} \to v_k$ as $n \to \infty$, and let $f_n = \sum_{k=1}^K v_{k,n} \chi_{A_k}$. Then $f_n \to f$ in $L^2((0,T),V)$.

Also, if $A$ is dense in $B$, and $B$ is dense in $C$, then $A$ is dense in $C$.

Hence the simple $V_h$ values functions are dense in $L^2((0,T),V)$.