Showing existence of a projective

Question

Let $X$ be a normed space and $Y$ be a finite dimensional subspace of $X$. Show that there is a projective $P\in B(X)$ such that $Im P=Y$.

Hint: First Solve for $dimY=1$ then generalize the solution for any finite dimensioanl spaces.

Here since $Y$is finite dimensional subspace of a normed space so it is closed. However we are not given that $X$ is banach, so we cannot use any theorem about projective in banach spaces!.

**The question stated in Question about proof that finite-dimensional subspaces of normed vector spaces are direct summands Suggests another way to show this.

Can someone kindly, explain how to solve the question using the hint.

Answer 1

Hint. Use the Hahn-Banach theorem. Note that it doesn't involve Banach spaces anyway.

So what about the $1$-dimensional case? You didn't write what you did with this hint.