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Let $X \neq \{0\}$ a normed vector space.Prove the following

(a) $X$ does not have isolated points.

(b) If $x,y \in X$ such that $ ||x-y||= \epsilon >0$ then

1.Exists a sequence $(y_n)_n$ in $X$ such that $||y_n-x|| < \epsilon \quad $ for all $n$ and $ y_n \to y$

2.Exists a sequence $(y'_n)_n$ in $X$ such that $||y'_n - x|| > \epsilon \quad $ for all $n$ and $y'_n \to y$.

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If it is a homework, put an appropriate tag. What did you try? –  Ilya Feb 22 '12 at 12:59
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1 Answer 1

Hints.

a. For $x\in X$ you can define $x_n = (1-\frac{1}{n})x$ and find $\|x-x_n\|$.

b1. consider $y_n = \alpha_n x+(1-\alpha_n)y$ with $\alpha_n\in(0,1)$ and $\alpha_n\to 0$ with $n\to\infty$.

I guess, for b2. you can imagine a similar example.

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Thank's for the reply! –  passenger Feb 22 '12 at 16:28
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