# Stupid Question

1. Oct 23, 2011

### LikeMath

I am sure that my questions are stupid. If we have a Hilbert space H, what do we mean by the closed subspace of H. Also, Does every Hilbert space have an identity? :P.

Could any one please clean to me these things .
Thanks!

2. Oct 23, 2011

### micromass

Staff Emeritus
A subspace of a vector space (and a Hilbert space is a vector space) is a nonempty set X such that

• $x,y\in X~\Rightarrow~x+y\in X$
• $x\in X,\alpha\in \mathbb{C}~\Rightarrow~\alpha x\in X$

A Hilbert space also comes equipped with a norm:

$$\|x\|=\sqrt{<x,x>}$$

and a set X is closed under the norm if for all convergent sequences in X it holds that the limit is in X.

A closed subspace is something that is both a subspace and closed.

What do you mean with identity?? It has a 0, which is the additive identity...

3. Oct 23, 2011

### LikeMath

Thank You.
I mean 1.

4. Oct 23, 2011

### micromass

Staff Emeritus
That doesn't really help me. What is 1 supposed to mean??

5. Oct 23, 2011

### LikeMath

1 is the multiplicative identity

6. Oct 23, 2011

### micromass

Staff Emeritus
And since when does a Hilbert space have a multiplication??

7. Oct 23, 2011

### LikeMath

Oops, thank you, that is why the question is stupid.

8. Oct 23, 2011

### micromass

Staff Emeritus
There are no stupid questions :tongue2: