# Infinite dimensional Hilbert Space

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?

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WWGD
Gold Member
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.

kent davidge
Mark44
Mentor
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.

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

kent davidge
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.

kent davidge
fresh_42
Mentor
2021 Award
If the space is infinite dimensional then it is not a manifold.
Why that?

kent davidge
martinbn