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Folland gives the following problem on page $159$ of his book Real Analysis:

Let $\mathcal X$ be a normed space.

If $\mathcal M$ is a closed subspace and $x\in \mathcal X \setminus \mathcal M$ then $\mathcal M + \mathbb C x$ is closed.

The only way I know how to do this is to consider the projection $\mathcal X \rightarrow \mathcal X/ \mathcal M$, note that the subspace generated by $x$ in the quotient space is closed, then note the inverse is exactly the space we want. The inverse is closed because it is the inverse of a closed set by a continuous map, so we're done.

However, the hint given by Folland asks the reader to use the fact that given a closed subspace $\mathcal M$ and point $x$ not in $\mathcal M$, there is a bounded functional $f$ that vanishes on $\mathcal M$ and is nonzero at $x$. (This is an easy consequence of Hahn-Banach.) How can this hint be used to solve the exercise?

This question is part a of the problem. The next part asks the reader to show that finite-dimensional spaces are closed, so I would like to avoid using that fact.

Christopher A. Wong
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Potato
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1 Answers1

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By multiplying the functional $f$ with a constant, we can assume $f(x) = 1$.

Then let $y \in \overline{\mathcal{M} + \mathbb{C}\cdot x}$. Let $\xi_n$ a sequence in $\mathcal{M} + \mathbb{C}\cdot x$ converging to $y$. Then $f(\xi_n) \to f(y)$, and that means the $\mathbb{C}x$ components of $\xi_n$ converge. Then $\xi_n - f(\xi_n)x \to y - f(y)x$ converges too, but $\xi_n - f(\xi_n)x \in \mathcal{M}$, so $y - f(y)x \in \mathcal{M}$, and $y = (y - f(y)x) + f(y)x \in \mathcal{M} + \mathbb{C}x$.

Daniel Fischer
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