I have problems in understanding the result in the following theorem(https://projecteuclid.org/download/pdf_1/euclid.pjm/1102993723). The $C_w(0,T;Y)$ means a $L^\infty(0,T;Y)$ function(a strongly measurable function with essential bound) a.e equal to weakly continuous function from closed interval $[0,T]$ to $Y$. The theorem seems to suggest intuitively that a weakly continuous function in a larger space with some essential bound in a smaller space is also weakly continuous with respect to the smaller space.The proof of it in this paper is:
The proof of this result seems not difficult. However I can not figure out where the density of $V$ in $Y$ is used. From the proofs it seems that $V$ is not needed to be dense in $Y$. I have been considering the proof for many times and fail to see why the density condition is necessary.Could any one help on this.