# Schwartz Distribution Theory in N variables?

• lugita15
In summary: Next question: What's a "diagonal linear functional"?In summary, the conversation discusses the possibility of extending Schwartz's theory of distributions to N variables and whether Schwartz's Kernel Theorem can be applied in this extension. The conversation also touches on the notion of distributional derivative in N variables and its relation to hyperfunctions. The use of Schwartz's Kernel Theorem for the distribution-valued inner product on a rigged Hilbert space is questioned, and the concept of "distributions over distributions" is explored. The conversation ends with a request for more information on constructing a two-dimensional delta function and a clarification on the terms "linear functional," "rigged Hilbert space," "bilinear functional," and "diagonal
lugita15
The title says it all - is it possible to extend Schwartz's theory of distributions to N variables? I've heard that it can be easily extended to two variables using Schwartz's Kernel Theorem (does anyone know a good reference that explains this stuff?). Does this theorem carry over to N variables? Also, how does the notion of distributional derivative generalize to N variables?

Finally, a largely unrelated question: I've seen assertions that hyperfunctions are "infinite order distributions." What does that even mean? What is the order of a distribution, and how do hyperfunctions relate to distribution theory?

Any help would be greatly appreciated.

lugita15 said:
does anyone know a good reference that explains [Schwartz's Kernel Theorem]
Haven't found what I'd call a "good" reference, but this one is certainly a classic reference:

Gelfand & Vilenkin (vol 4 of a series by Gelfand and several collaborators).

strangerep said:
Haven't found what I'd call a "good" reference, but this one is certainly a classic reference:

Gelfand & Vilenkin (vol 4 of a series by Gelfand and several collaborators).
Speaking of Gelfand Volume 4, is Schwartz's Kernel Theorem used for the distribution-valued inner product on a rigged Hilbert space?

lugita15 said:
Speaking of Gelfand Volume 4, is Schwartz's Kernel Theorem used for the distribution-valued inner product on a rigged Hilbert space?
Not directly, afaik. But it occurs quite early in G&V (iirc), even before they start to construct nuclear spaces and their duals.

strangerep said:
Not directly, afaik. But it occurs quite early in G&V (iirc), even before they start to construct nuclear spaces and their duals.
But there should be some dependence, right, at least indirectly? Because the inner product is a delta function of two variables, and distributions of two variables require Schwartz's Kernel theorem.

By the way, I asked you about this in my RHS thread, but the distribution-valued inner product is a distribution on the rigged Hilbert space, so does that mean you need to define test functions on the rigged Hilbert space, and the inner product will be some kind of functional acting on those test functions, just like the plain-old Schwartz theory for ordinary spaces? Since the rigged Hilbert space (that is, the ket space) is anyway made up of distributions, we're talking about "distributions over distributions" as opposed to "distributions over numbers"!

You can always apply multi-linear algebra...

If I recall correctly, the tensor product of the space of test functions in M variables with the space of test functions in N variables turns out to be the space of test functions in M+N variables, so this approach works out particularly nicely.

lugita15 said:
But there should be some dependence, right, at least indirectly? Because the inner product is a delta function of two variables, and distributions of two variables require Schwartz's Kernel theorem.
Er,... no. A delta function ##\delta(k-p)## can be regarded as a family of ordinary distributions ##\delta_p##, for each of which we have
$$\Big(\delta_p \,,\, f\Big) ~=~ f(p)$$
where the large parentheses denote the dual pairing operation.

By the way, I asked you about this in my RHS thread, but the distribution-valued inner product is a distribution on the rigged Hilbert space, so does that mean you need to define test functions on the rigged Hilbert space, and the inner product will be some kind of functional acting on those test functions, just like the plain-old Schwartz theory for ordinary spaces? Since the rigged Hilbert space (that is, the ket space) is anyway made up of distributions, we're talking about "distributions over distributions" as opposed to "distributions over numbers"!
Hmmm. Almost everything you said in that paragraph is wrong, or distorted. I don't know where to start...

Let's put distribution-valued inner products aside for a while.

Tell me which texts on rigged Hilbert space you've actually studied in detail, and then tell me what you think a "linear functional" is, what a "rigged Hilbert space" is, and what a "bilinear functional" is.

Hurkyl said:
You can always apply multi-linear algebra...

If I recall correctly, the tensor product of the space of test functions in M variables with the space of test functions in N variables turns out to be the space of test functions in M+N variables, so this approach works out particularly nicely.

Yes, but the step implied in the above, i.e.,
$$(V \otimes V)^* ~=~ V^* \otimes V^*$$
is straightforward in the finite-dim case but tricky in the inf-dim case -- which is what the SKT covers, iiuc.

strangerep said:
Er,... no. A delta function ##\delta(k-p)## can be regarded as a family of ordinary distributions ##\delta_p##, for each of which we have
$$\Big(\delta_p \,,\, f\Big) ~=~ f(p)$$
where the large parentheses denote the dual pairing operation.
Where can I find more information about constructing a two-dimensional delta function as a family of delta functions? Would it be there in Gelfand?
strangerep said:
Tell me which texts on rigged Hilbert space you've actually studied in detail
Just various bits and pieces gathered from Gelfand vol 4, Rafael de la Madrid's various papers, and PhysicsForums. Nothing in real detail.
strangerep said:
and then tell me what you think a "linear functional" is, what a "rigged Hilbert space" is, and what a "bilinear functional" is.
A linear functional is a linear mapping from a vector space to the real line, and a bilinear functional is a linear mapping from the Cartesian product of a vector space and itself to the real line. A rigged Hilbert space arises from the Gelfand triplet consisting of a Hilbert space, a nuclear subspace on which unbounded Hilbert space operators are defined, and a space of anti-linear functionals, essentially Schwartz distributions except for the anti-linear part, using the nuclear subspace as the set of test functions on which the functionals act. But you'll have to excuse me, because I often slip up and just refer to the ket space, i.e. the space of anti-linerar functionals, out of this triplet as the "Rigged Hilbert Space".

lugita15 said:
A linear functional is a linear mapping from a vector space to the real line, and a bilinear functional is a linear mapping from the Cartesian product of a vector space and itself to the real line.
OK, that's reasonably close. The actual definitions are a little more general, cf.
http://en.wikipedia.org/wiki/Bilinear_functional
Also, for QM we usually take the range to be the complex plane rather than just the real line.

A rigged Hilbert space arises from the Gelfand triplet consisting of a Hilbert space, a nuclear subspace on which unbounded Hilbert space operators are defined, and a space of anti-linear functionals, essentially Schwartz distributions except for the anti-linear part, using the nuclear subspace as the set of test functions on which the functionals act.
The case of Schwartz test functions and tempered distributions is a special case. But ok, that's enough to go on with...

[...] I often [...] just refer to the ket space, i.e. the space of anti-linear functionals, out of this triplet as the "Rigged Hilbert Space".
IMHO, that's actually a reasonable complaint. There's no overarching term for the whole thing, afaik. Maybe it deserves to be called a "Dirac space", since it was his approach to QM that motivated much of the rigorous development. :-)

Where can I find more information about constructing a two-dimensional delta function as a family of delta functions? Would it be there in Gelfand?
It depends what you mean by "two-dimensional delta function". I interpret it to mean a
bilinear functional ##\delta_{pq},~ p,q \in \mathbb{C}##, acting on some vector space ##\mathbb{F}## of functions over ##\mathbb{C}^2##, and defined via
$$\def\Cz{\mathbb{C}} \def\Fz{\mathbb{F}} \Big( \delta_{pq} \,,\, f \Big) ~=~ f(p,q)$$
Normally, we'd like to write the above as
$$\int dx \delta(p-x) \int dy \, \delta(q-y) \; f(x,y) ~=~ f(p,q) ~.$$

But now consider an arbitrary bilinear functional K satisfying
$$\Big( K \,,\, f \Big) ~=~ z_f$$
where ##z_f## is a family of complex numbers dependent on the function f.

The Schwartz kernel theorem (iiuc!) states that one can always express ##K##
and its action in the form
$$\int dx \int dy \, K(x,y) \; f(x,y) ~=~ z_f ~.$$
That's why it's called the "kernel" theorem. The quantity ##K(x,y)## is the "integral kernel" defining the bilinear functional.

Depending on the details of K, it may or may not be possible to express it as a simple product ##A(x)Y(y)## as we did for the 2D delta distribution.

So maybe you can now re-study the sections in G+V on the kernel theorem, and you (or anyone else -- please!) can tell me whether I've got it right... :-)

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## What is Schwartz Distribution Theory in N variables?

Schwartz Distribution Theory in N variables is a mathematical theory that extends the concept of a distribution (a generalized function) to multiple variables. It provides a rigorous framework for dealing with functions that are not necessarily continuous or differentiable, and is commonly used in fields such as physics and engineering.

## What are some key concepts in Schwartz Distribution Theory in N variables?

Some key concepts in Schwartz Distribution Theory in N variables include the notion of a test function, which is a smooth function with compact support, and the duality pairing between distributions and test functions. The theory also includes the concept of convergence of distributions and operations such as differentiation and multiplication by smooth functions.

## How is Schwartz Distribution Theory in N variables useful?

Schwartz Distribution Theory in N variables is useful in solving problems in which traditional methods fail due to the presence of discontinuities or singularities. It allows for the manipulation and analysis of generalized functions, which can model physical phenomena more accurately than traditional functions. It also provides a framework for the study of partial differential equations and their solutions.

## What are some applications of Schwartz Distribution Theory in N variables?

Schwartz Distribution Theory in N variables has many applications in fields such as mathematical physics, engineering, and signal processing. It is used to solve differential equations, analyze signals and systems, and model physical phenomena such as shock waves and quantum mechanics. It is also used in the study of wavelets and Fourier analysis.

## What are some limitations of Schwartz Distribution Theory in N variables?

Although Schwartz Distribution Theory in N variables is a powerful mathematical tool, it has some limitations. For example, it cannot be applied to distributions with unbounded support, and certain operations such as division are not well-defined. Additionally, it can be difficult to find explicit solutions to differential equations involving distributions, as the theory often deals with generalized solutions rather than exact ones.

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