Infinite dimensional Hilbert Space

[kent davidge]

Could someone tell me in what sense the following photo of Hilbert is a infinite dimensional Hilbert Space?

It's shown in a pdf I'm reading.

Perhaps I'm putting the chariot in front of the horses as one would say here in our country, by considering infinite as infinite dimensional?

 

[WWGD]

Are you sure this is meant literally? It may be a joke comment. A Hilbert space is a metric space. I don't see any mention , neither implicit nor explicit of any metric.

 

[Mark44]

A Hilbert space can be finite-dimensional or infinite-dimensional. The objects in an infinite-dimensional Hilbert space are infinite sequences, and are considered to be infinite-dimensional vectors.

The caption under the picture isn't a Hilbert space, obviously -- I believe it is merely commenting on what is probably his most well-known work.

 

[martinbn]

May be if you show pictures (a), (b), ..., (o) and the context of all this would be better.

 

[kent davidge]

Here is the pdf https://web.stanford.edu/~jchw/WOMPtalk-Manifolds.pdf

 

[martinbn]

Well, you can safely ignore that bit, these notes are not about Hilbert spaces. The picture is of course Hilbert himself, not a Hilbert space, perhaps it is supposed to be witty. If the space is infinite dimensional then it is not a manifold. I think that is the point to realize.

 

[fresh_42]

If the space is infinite dimensional then it is not a manifold.

Why that?

 

[martinbn]

Yes, but the notes from the link consider only manifolds that a locally $\mathbb R^n$.

 

[WWGD]

Yes, there are Banach manifolds too, but it font know enough about them.