Does every Hilbert space have an identity?

[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 anyone please clean to me these things .
Thanks!

 

[micromass]

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.

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.

Also, Does every Hilbert space have an identity? :P.

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

 

[LikeMath]

Thank You.

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

I mean 1.

 

[micromass]

I mean 1.

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

 

[LikeMath]

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

1 is the multiplicative identity

 

[micromass]

1 is the multiplicative identity

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

 

[LikeMath]

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

 

[micromass]

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

There are no stupid questions