Let $V$, a finite dimensional vector space, and $L$, a subspace of $V$. Let $T:V^*\rightarrow L^*$ defined as: $T(\varphi)(x)=\varphi(x)$ for all $\varphi \in V^*$. Prove $T$ is onto.
Well, I'm struggling with understanding what are $L^*, V^*$ with relation to $L, V$.
Do you have an explanation for that?
$V^*$ is the dual space to $V$, that is the vector sapce of all linear maps $V\to k$ to the ground field (this is a vector space via pointwise addition and mutlpplication with scalars). So if $\phi\in V^*$ is a linear map $V\to k$, $v\mapsto \phi(v)$, then $T(\phi)$ is the linear map $L\to k$ given by $x\mapsto \phi(x)$, i.e. the restriction to $L$.
Remark: Unless you have issues with the Axiom of Choice, the condition that $V$ be finite dimensional can be dropped.