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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.

user652838
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    If you really want to use Banach spaces, can you not embed $X$ in its completion $\hat{X}$? The embedding won't be closed, but since $Y$ is finite-dimensional rather than a general closed subspace, passing to $\hat{X}$ doesn't change anything... – Brian Moehring Apr 26 '21 at 18:27
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    I don't think involving Banach spaces will be useful. This isn't a question about completeness. In a Hilbert space, you could use an orthonormal basis though. – J. De Ro Apr 26 '21 at 18:30
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    If you read that proof carefully, they are using the one dimensional case to look for those $\alpha_i$. – Arctic Char Jul 02 '21 at 15:01
  • I did not much understand what you mean..They said that the other question solves my question, I don't see how.@Arctic Char – user652838 Jul 03 '21 at 06:59

1 Answers1

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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.

Michał Miśkiewicz
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    It's not the right place to explain another question, but let me just say that $x \mapsto \alpha_1(x)e_1 +\ldots+ \alpha_n(x)e_n$ is the required projection, where $\alpha_j$'s are extended to the whole space. – Michał Miśkiewicz Jun 08 '21 at 09:13