# Injection for the dualspace of a subspace into the dualspace of the entire space

Assume $V$ is a finite dimensional vectorspace and let $W$ be a subspace of $V$. Denote by $i \colon W \to V$ the injection of $W$ into $V$.

Then it is possible to define a dual map $\pi \colon V^* \to W^*$, the projection from the dualspace of $V$ to the dualspace of $W$.

My question is now, if there is a canonical way of defining a map $\lambda \colon W^* \to V^*$ as the embedding from $W^*$ to $V^*$

Not without extra data. There are many injective linear maps $\lambda \colon W^{*} \rightarrow V^{*}$. Even if you require compatability with $\pi$ in the sense that $\pi \circ \lambda = \operatorname{id}|_{W^{*}}$, then such an injective linear map provides a "linear scheme" of extending functionals that are defined on $W$ to functionals that are defined on $V$ and there are many possible ways to do this (by choosing a basis for $W$, extending it to a basis of $V$ and letting the functional act arbitrary on the extra basis elements).
If you choose a complement $W'$ to $W$ so that $V = W \oplus W'$, then you have a surjective projection $\pi_{W|W'} \colon V \rightarrow W$ to $W$ along $W'$ (that depends on both $W$ and $W'$!) and then the dual map $\pi_{W|W'}^{*} \colon W^{*} \rightarrow V^{*}$ will give you an injective linear map from $W^{*}$ to $V^{*}$. If you are working over $\mathbb{R}$ or $\mathbb{C}$ and have an inner product, you can take $W' = W^{\perp}$.