Let $W$ a subspace of $V$. Show that every element of $W^{*}$ is the restriction of an element of $V^{*}$ to $W$.

Question

The question is:

Let $W$ a subspace of $V$. Show that every element of $W^{*}$ is the restriction of an element of $V^{*}$ to $W$.

I think that this question is true only if $\dim(V)<\infty$, cause we could use the fact that for every basis $\alpha$ for $W$, exists a dual basis $\alpha^{*}$ for $W^{*}$. But if $\dim(V)=\infty$, the set $\alpha^{*}$ is linearly independent but not necessarily a basis.

Answer 1

If there is no norm or topology given, $V^*$ denotes the algebraic dual $\hom(V, K)$ where $K$ is the field of scalars.
Then, using a form of axiom of choice (e.g. transfinite recursion), we can fix a basis $(w_i)$ for $W$, and extend it by elements $(v_j)$ to obtain a basis of $V$.

Then any linear $\alpha:W\to K$ can be extended to $V\to K$ by assigning arbitrary values (e.g. $0$) to the $v_j$'s.

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If the infinite dimensional space $V$ is equipped with a norm, $V^*$ usually denotes the topological dual $\{\alpha:V\to K\mid\alpha$ is linear and continuous$\}$, but then we can use the Hahn-Banach theorem.