# functional analysis-normed linear space

Let $V$ be a norm linear space and let $x\in V\setminus\{0\}$. Also let $W$ be a linear subspace of $V$. Show that if there is $r>0$ such that $\{y \in V \mid \lVert y\rVert< r\}$ is a subset of $W$, then $\frac{rx}{2\lVert x\rVert}\in W$.

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I have edited the question. However I am not sure what exactly was your question at the end. Please use LaTeX in the future and try and ask questions with a bit more clarity in your words.It'll help you get better answers. –  Vishesh Oct 21 '12 at 14:51
Right, Davide got it clean. Thanks. –  Vishesh Oct 21 '12 at 14:53

Let $B_r=\{y\in V: ||y||<r\}$. Because $||x||\ne 0$, we get $$||\frac{r}{2||x||}x||=|\frac{r}{2||x||}|\cdot ||x||=\frac{r}{2||x||}\cdot ||x||=\frac{r}{2}<r.$$ This implies that $\displaystyle{\frac{r}{2||x||}x}\in B_r$ and because $B_r\subseteq W$, the result follows. Thats what Davide means...
Hint: what is the norm of $\frac r{2\lVert x\rVert}x$? What can we say about elements which have norm $<r$?