I want to prove (or actually find a counter-example) of the following problem, regarding Browder's theorem. Let $X$ be a reflexive, separable Banach space and $Y$ a finite-dimensional subspace. Furthermore, be $B \colon X \to X^*$ demicontinuous and bounded (i.e. maps bounded sets to bounded sets). Then follows the continuity of the mapping \begin{align*} \tilde{B} \colon Y &\to X^* \\ u &\mapsto Bu. \end{align*}
I've got some concerns, because $X^*$ is not even embedded in $Y^*$. Can someone help me? Thanks.