Is adjoint operator a projection?

Question

Let $X$ be a reflexive Banach space and $Y$ its finite dimensional subspace. Let $i_Y\in \mathcal{L}(Y,X)$ be the embedding operator. Why $i^*_Y\in \mathcal{L}(X^*,Y^*)$ is the projection on $Y^*$?

Answer 1

Recall that adjoints are defined in the following manner: for any functional $\ell \in X^*$, $$i_Y^*(\ell)=\ell \circ i_Y$$ where $\circ$ is function composition.

Now, for $y \in Y \subset X$, $i_Y(y) = y$, so $i_Y^*(\ell)(y) = \ell \circ i_Y(y) = \ell(y)$.

Thus, $i_Y^*$ is a "projection onto $Y^*$" in the sense that, when acting on a functional $\ell$, it gives the restriction of $\ell$ to the subspace $Y \subset X$, which is an element of $Y^*$.