# Norms in vector spaces

1. Jan 17, 2010

### Niles

Hi guys

The norm is a continuous function on its vector space, but I a little unsure of how to interpret this. Does it mean that if we are in e.g. a Hilbert space with an orthonormal basis ei (i is a positive integer), then we have

$$\left\| {\mathop {\lim }\limits_{N \to \infty } \sum\limits_{i = 1}^N {x_i e_i } } \right\| =\mathop {\lim }\limits_{N \to \infty } \left\| {\sum\limits_{i = 1}^N {x_i e_i } } \right\|$$

for some vector x = Σ xiei in the Hilbert space? Or does the above operation come from the continuity of taking limits?

2. Jan 17, 2010

### Landau

Yes, it does.
I'm not sure what you mean by this. Maybe this will clear things up:

You know that for a continuous function f and a converging sequence x_n, we have
$$\lim_{n\to\infty}f(x_n)=f(\lim_{n\to\infty}x_n)$$,
i.e. you can "pull the limit out of the argument of a continous function".

Now, let f be the norm-function: $$f=\|...\|$$. Then the above becomes
$$\lim_{n\to\infty}\|x_n\|=\|\lim_{n\to\infty}x_n\|$$
i.e. you can "pull the limit out of the (argument of) the norm", which is what you did.