Infinite dimensional Hilbert Space

  • #1
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Could someone tell me in what sense the following photo of Hilbert is a infinite dimensional Hilbert Space?
xNNpZry.png


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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  • #2
WWGD
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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.
 
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  • #3
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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.
 
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  • #4
martinbn
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May be if you show pictures (a), (b), ..., (o) and the context of all this would be better.
 
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  • #6
martinbn
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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.
 
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  • #8
martinbn
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Yes, but the notes from the link consider only manifolds that a locally ##\mathbb R^n##.
 
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  • #9
WWGD
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Yes, there are Banach manifolds too, but it font know enough about them.
 
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