Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $y'_n \to y$.

share|cite|improve this question

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.

share|cite|improve this answer
    
Thank's for the reply! – passenger Feb 22 '12 at 16:28

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.