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necessary and sufficient. This shouldn't be too hard for somebody reasonably knowledgeable in the subject. It is completely unnecessary and in fact needlessly confusing to define here what is meant by "finer topology" when there is a link to that already. And please don't ramble on in the middle of a formal definition as to why or how this or that concept or theorem in connected. (All that's important and should remain part of the article, but not as part of a definition.) And don't "consider" things or use vague language such as "of some sort" in the middle of a definition, either.
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695:"that is one for which the natural inclusion" (why "natural"?) "are faithfully represented as distributions (because we assume Φ dense)" (why "because"?) "the definition of rigged Hilbert space is in terms of a sandwich" (what is a "sandwich"?) "a simple example is given by Sobolev spaces" (why is this "simple"?)
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Shouldn't the last sentence of the article: " Note that even though Φ is isomorphic to Φ* if it happens that Φ is a
Hilbert space in its own right, this isomorphism is not the same as the composition of the inclusion i with its adjoint i*" read "Note that even when Φ is isomorphic to Φ* so that Φ is
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But how does this interact with the L^2 structure of L^2? For example, how does it interact with the existence of an L^2 eigenbasis of "bound states" for an elliptic operator, or compact operator, or
Fredholm operator somewhere? More ambitiously, how does it interact with approximate eigenvectors and
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I don't really know what a rigged
Hilbert space is. It seems to be some extension of a Hilbert space, but that is all I got from reading this article. A lot of important mathematical concepts are mentioned quite casually in the article, but a true formal definition is missing. When I see the words
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I have a Ph.D. in mathematics (analysis) and I find this article hard to read. I think it simply needs to be longer with longer explanations. I have seen a better explanation in a quantum mechanics book. Such phrases as "Phi 'carries' a finer topology", and "linear functionals on the subspace Phi
311:, not just functions. This concept is capable of expressing that. It pays dividends in spectral theory, for example, where it can be more interesting to know more about eigenvectors than just that they are vectors. They are typically functions or distributions, with smoothness and other properties.
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Sorry if this sounds like a rant, but I am a mathematical type person, not a physicist, and I really do have trouble understanding any kind of mathematical concept without a formal definition. Perhaps someone will be so kind as to include a formal definition in this article or put a little box on
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This article seems to have two definitions of a rigged
Hilbert space, neither of them formal. It would be better to say something on the order of: "A rigged Hilbert space is a Hilbert space H together with a subspace Φ ... such that:" followed by a formal list of conditions which together are
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I don't see a nice way of inserting the definition into the article without restructuring it significantly, and the technical definition is actually there among all the admittedly long winded verbiage. The definition is, a rigged
Hilbert space is a triple (H,S,S') of a Hilbert space, a dense
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for v in H are 'faithfully represented', etc." may cause trouble even for mathematicians. The "Formal definition (Gelfand triple)" section is better, but there should be at least a link with "the adjoint to i'. The assertion that "this isomorphism is not the same and the composition of the
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Also, the article says that a rigged
Hilbert space is a "pair". Yet it then goes on to talk about a "triple". I cannot seem to understand what the relationship is between this pair and this triple. For example, what is the point of this triple? It is actually necessary?
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I suppose you can take Φ to be C^∞ functions of compact support, Φ* to be distributions, and then we have Φ ⊆ L^2 ⊆ Φ*. Then L := i d/dx is defined from Φ* to itself, and u := exp(iλx) satisfies Lu = -λu in this space, so it is an eigenvector of L:Φ* → Φ*.
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Exactly. In the
Gelfand-style approach there is the extra 'degree of freedom' this allows (choose Φ to fit the problem). Connected certainly with the ideas of doing representation theory in infinite-dimensional spaces. With a Lie group acting, say in a
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that its elements can represent, say square-integrable holomorphic functions on the unit disc, than just to know about a definition in terms of square-summable sequences. Equally it is interesting to know about the elements of
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Finally, what is the purpose of this statement? Is it necessary? "if it happens that Φ is a
Hilbert space in its own right, this isomorphism is not the same as the composition of the inclusion i with its adjoint i*"
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this article that says "in need of expert attention." (I don't know how to do either one.) But I do know a formal mathematical definition when I see one, and this article doesn't have one, and it really needs one.
195:"a rigged Hilbert space is," or "Formally, a rigged Hilbert space consists of ..." I expect to see a formal mathematical definition, not a bunch of advanced mathematical concepts casually thrown together.
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As pointed out before, this article is very technical and thus hard to understand. And I say this as a PhD in theoretical physics. In places, the language also comes across as patronising to the reader.
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Such spaces were introduced to study spectral theory in the broad sense. They bring together the 'bound state' (eigenvector) and 'continuous spectrum', in one place.
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several times now, though I am quite sure those are not canonically isomorphic, while still isomorphic as vector spaces. I'd suggest to change the notation to
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Term "rigged" should be interpreted as "equipped and ready for action", in analogy with the rigging of a sailing ship. -
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point spectrum/continuous spectrum/residual spectrum of a (possibly unbounded) operator acting on L^2?
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subspace (in the strong top), and its dual, viewed as a superset of H by means of Riesz. HTH -
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My difficulty with this article is the non-fulfillment of the promise made in the beginning:
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inclusion i with its adjoint i^*..." should be backed up with a reference or counterexample.
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it is potentially confusing: it is given by a canonical injection (monomorphism), not an
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I assumed it meant "artificially constructed", as in, "this poker game is rigged". :-)
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The topology on the subspace makes it into a locally convex space.--
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