Subspace of Normed Vector Space

**iknowone** · September 10th 2009, 10:32 AM

Let X be a normed vector space. If C is a closed subspace x is a point in X not in C, show that the set C+Fx is closed. (F is the underlying field of the vector space).

**Opalg** · September 10th 2009, 01:57 PM

Originally Posted by iknowone

Let X be a normed vector space. If C is a closed subspace x is a point in X not in C, show that the set C+Fx is closed. (F is the underlying field of the vector space).

Let $d = \inf\{\|x-c\|:c\in C\}$ be the distance from x to C. Since C is closed, d>0. For $c\in C$ and $0\ne\lambda\in F$ , $\|c+\lambda x\| = |\lambda|\|\lambda^{-1}c +x\|\geqslant d|\lambda|$ , so that $|\lambda|\leqslant d^{-1}\|c+\lambda x\|$ . That shows that the map $\phi:c+\lambda x\mapsto\lambda$ is continuous from C+Fx to F.

Now suppose that $(c_n+\lambda_n x)$ is a convergent sequence in C+Fx, with $\textstyle\lim_{n\to\infty}c_n+\lambda_n x = y\in X$ . (We want to show that $y\in C+Fx$ .) Then $c_m-c_n + (\lambda_m-\lambda_n)x\to0$ as $m,n\to\infty$ . Deduce from the continuity of $\phi$ that $(\lambda_n)$ is Cauchy in F, hence converges to $\lambda_0$ say (presumably the scalar field F is meant to be complete). It then follows that $c_n\to y-\lambda x = c_0$ say, as $n\to\infty$ . But C is closed, so it follows that $c_0\in C$ . Finally, $c_n+\lambda_n x \to c_0+\lambda_0x$ , so $y = c_0+\lambda_0x \in C+Fx$ as required.

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