Subspaces and convergence in weak* topology

Question

I would like to ask some questions regarding convergence in the weak* topology and subspaces.

Let $X$ be a normed space with subspace $A \subset X$. Assume $X$ is endowed with the weak* topology. Let $\{u_{m}\}_{m}$ be a sequence in $X$.

1. If $u_{m} \rightharpoonup^{*} u$ in $A$ then since $X$ is not necessarily first-countable in the weak* topology then would I be right in stating that $u$ is not necessarily in $A$?

2. If $u_{m} \rightharpoonup^{*} u$ in $X$ then does that imply that $u_{m} \rightharpoonup^{*} u$ in $A$ if $\{u_{m}\}_{m} \subset A$? Similarly, if $u_{m} \rightharpoonup^{*} u$ then does that imply that $u_{m} \rightharpoonup^{*} u$ in $X$?

3. Would question 2 require that $A$ is closed in $X$?

I am trying to apply this to Sobolev space $W^{1,p}(\Omega)$ and closed subspace $W^{1,p}_{0}(\Omega)$. Any assistance or hints would be appreciated.

Answer 1

1. No. You assume that $u_n\overset{w^*}{\to} u$ in $A$ and then asks if $u\in A$. Of course, $u\in A$, because of the assumption.

2. Yes. See this answer.

3. No. Carefully read the proof given in the link.