# Extending to closed space

• Nov 28th 2009, 02:40 PM
putnam120
Extending to closed space
Let $\displaystyle X$ be a normed vector space and $\displaystyle M\subseteq X$ a closed subspace. Show that $\displaystyle M+\mathbb{C}x$ is also closed.

I was told to try using the Hahn-Banach theorem. So I know that there exists a function $\displaystyle f\in X^{*}$ such that $\displaystyle \parallel f\parallel =1$ and $\displaystyle f(x)=\delta$. Where $\displaystyle x\in X\cap M^c$ and $\displaystyle \delta=\inf_{y\in M}\parallel x-y\parallel$

However, I don't see how to use this at all.
• Nov 29th 2009, 12:17 AM
Opalg
Quote:

Originally Posted by putnam120
Let $\displaystyle X$ be a normed vector space and $\displaystyle M\subseteq X$ a closed subspace. Show that $\displaystyle M+\mathbb{C}x$ is also closed.

I was told to try using the Hahn-Banach theorem. So I know that there exists a function $\displaystyle f\in X^{*}$ such that $\displaystyle \parallel f\parallel =1$ and $\displaystyle f(x)=\delta$. Where $\displaystyle x\in X\cap M^c$ and $\displaystyle \delta=\inf_{y\in M}\parallel x-y\parallel$

However, I don't see how to use this at all.

To show that $\displaystyle M+\mathbb{C}x$ is closed, suppose that you have a convergent sequence $\displaystyle m_k+\lambda_kx\to y$. You need to show that the limit point y is also in $\displaystyle M+\mathbb{C}x$. From the continuity of your function f above, $\displaystyle \lambda_k\delta = f(m_k+\lambda_kx)\to f(y)$ as $\displaystyle k\to\infty$. Deduce that $\displaystyle m_k\to y-\delta^{-1}f(y)x$. Since M is closed, it follows that $\displaystyle z = y-\delta^{-1}f(y)x\in M$, so that $\displaystyle y = z + \delta^{-1}f(y)x\in M+\mathbb{C}x$.
• Nov 29th 2009, 03:10 PM
putnam120
Thanks. In the time between our post I had figured out that I would have to show that a sequence of that form converged to a point in M. However, I was stuck at that part. (I was thinking about using the fact that the bounded linear functionals separate points somehow)