Are there ways to embed a Hilbert space H into a larger space G (not necessarily a Hilbert space) so that G is neither a direct sum of H with another nor a tensor product of H with another? If the answer is yes, then some details on this would be much appreciated, thank you!
Look at
https://math.virginia.edu/seminars/colloq/files/Gilles_Fall2020.pdf
Thank you. I'll look at this in detail, soon. Then I'll probably come back to you with questions!
In addition to direct sums and tensor products, there are other ways in which a Hilbert space can be extended. Here are a few examples:
Orthogonal sum: An orthogonal sum of Hilbert spaces is a way to combine them into a new Hilbert space such that each component space is orthogonal to the others. This can be thought of as a generalization of the direct sum, where the component spaces need not be finite-dimensional.
Completion: Given a pre-Hilbert space (a vector space equipped with an inner product that is not necessarily complete), we can complete it to obtain a Hilbert space. The resulting space is the smallest Hilbert space containing the pre-Hilbert space.
Induced representation: Given a group acting on a Hilbert space, we can construct a new Hilbert space by taking the space of functions on the group that satisfy certain properties. This new space, called the induced representation, can be thought of as an extension of the original Hilbert space.
Direct integral: A direct integral of Hilbert spaces is a way to combine them into a new Hilbert space using a measure on a suitable space of parameters. This construction generalizes the direct sum to an infinite-dimensional setting and can be used, for example, to construct the space of square-integrable functions on a manifold.
These are just a few examples of ways in which a Hilbert space can be extended beyond a direct sum or tensor product. There are many other ways to construct new Hilbert spaces using various mathematical tools and techniques
Dear Yazen, thank you for your thoughts. Unfortunately I left my question slightly vague, but when I say `direct sum' I mean to refer to both orthogonal sums and and direct integrals too.
Regarding `completion': I assume a complete Hilbert space H from the outset. I suppose one could define a new inner-product on HxH so that H is pre-Hilbert space with respect to this new inner-product, then complete to get a new Hilbert space G, but then H is dense in G. I apologise for my question being a bit vague, but this is not what I was looking for.
However, the `induced representation' is potentially more interesting. But Is the new Hilbert space not a tensor product?
kind regards.
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