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Thread: Closed linear manifold

  1. #1 Closed linear manifold 
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    So, I am sitting waiting for my eye appointment, and idly reading Halmos on Hilbert spaces (strange looks from my neighbours!).

    I said "idly" as the very early stuff was quite familiar, but then I get to this, which is almost (not quite) verbatim.

    Suppose that be an Hilbert space (usual definition), and that is a subset. will be a linear manifold if, for all and any that .

    Fair enough, we have a definition. But now....

    If is a closed linear manifold then it is a {vector} subspace of .

    OK, I know the general definition of a manifold, and I know what the closed subsets of a topological space are, but I have never come across the notion of a closed manifold, linear or otherwise.

    And why does this make it a vector subspace? I am sure I am being dim here, but any help would be welcomed.

    PS There are some obvious questions that arise from this construction: Like, does this imply that every Hilbert space is a manifold, closed, linear or otherwise? Some are quite trivially manifolds in the "naive" sense (like and for any , for example), but is it generally true that all Hilbert spaces are closed linear manifolds, whatever that means?


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  3. #2 Re: Closed linear manifold 
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    Quote Originally Posted by Guitarist
    So, I am sitting waiting for my eye appointment, and idly reading Halmos on Hilbert spaces (strange looks from my neighbours!).

    I said "idly" as the very early stuff was quite familiar, but then I get to this, which is almost (not quite) verbatim.

    Suppose that be an Hilbert space (usual definition), and that is a subset. will be a linear manifold if, for all and any that .

    Fair enough, we have a definition. But now....

    If is a closed linear manifold then it is a {vector} subspace of .

    OK, I know the general definition of a manifold, and I know what the closed subsets of a topological space are, but I have never come across the notion of a closed manifold, linear or otherwise.

    And why does this make it a vector subspace? I am sure I am being dim here, but any help would be welcomed.

    PS There are some obvious questions that arise from this construction: Like, does this imply that every Hilbert space is a manifold, closed, linear or otherwise? Some are quite trivially manifolds in the "naive" sense (like and for any , for example), but is it generally true that all Hilbert spaces are closed linear manifolds, whatever that means?
    Just take it as terminology. It is a vector subspace. It has nothing to do with the meaning of "manifold" from topology and geometry.

    Halmos's terminology is a bit non-standard. He is using "linear manifold" for the usual algebraic meaning of subspace, and reserving "subspace" for those which are also topologically closed. More typically people just say "subspace" and "closed subspace".

    Terminology in mathematics is not rigid, so long as the author clearly defines his terms.


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  4. #3  
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    "Closed" in this context usually means all limit points are included - remember that Hilbert space is infinite dimensional.
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  5. #4  
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    Quote Originally Posted by mathman
    "Closed" in this context usually means all limit points are included - remember that Hilbert space is infinite dimensional.
    Hilbert spaces are not necessarily infinite-dimensional.

    A Hilbert space is a real or complex topological vector space in which the topology is induced by a skew-symmetric positive-definite inner product, and which is complete in the induced metric.

    Ordinary n-spce with the usual "dot product" is a Hilbert space.

    "Closed" means topologically closed. For a metric space sequentially closed implies closed.

    To say that "closed means all limit points are included" begs the question as to what is a limit point. The usual definition of a limit point of a set S is a point p such that every neighborhood of p contains a point of S. With that definition S is closed if and only if every limit point of S belongs to S and it does not matter what kind of topological space S is. If you demand that a limit point be a limit of a sequence of points of S then the situation is somewhat different in general, but in a metric space any limit point of S in the topological sense is a limit of a sequence of points in S.
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  6. #5  
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    Quote Originally Posted by DrRocket
    A Hilbert space is a real or complex topological vector space in which the topology is induced by a skew-symmetric positive-definite inner product, and which is complete in the induced metric.
    At the risk of being accused of sycophancy, I would say this is a masterfully succinct definition of an Hilbert space.

    I would note that, for some Hilbert spaces, like with the Euclidean metric the topology is easily found, as is completeness; for others it is a bit more tricky.

    These latter are the ones that Paul Halmos (and others, perhaps including those here?) is helping me with. Stay tuned for more questions (or not, your choice)
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  7. #6  
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    In my student days Hilbert space was the term used for infinite dimensional space, while Euclidean was the term for finite dimensional.

    Before any carping, I am talking about inner product spaces with the usual norm definition and also completeness (needed for infinite dimensional case).
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