Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I find it a bit strange that if $x_n \to x$ in Banach space $X$, then $|x_n|_X \to |x|_X$ by inverse triangle inequality... Surely that can't be right.

Am I correct in this? Does this have a name? Thanks

share|improve this question

1 Answer 1

up vote 3 down vote accepted

This is just the fact that the norm is a continuous function on a normed linear space.

share|improve this answer
    
Some people think it's necessary to prove that it's continuous under the typical metric space definitions. In this case, you can show that: Assume $||x_n - a|| \rightarrow 0$ Then $||x_n|| - ||a|| \leq ||x_n - a||$ and also $||a|| - ||x_n|| \leq ||x_n - a||$ and so $| ||x_n|| - ||a|| | \rightarrow 0$ –  muzzlator Feb 4 '13 at 15:06
    
And that is the correct proof. –  ncmathsadist Feb 4 '13 at 21:20

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.