Stupid Question

[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!

[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 :tongue2: