Schwartz Space and Generalized Functions

Schwartz Space

Schwartz Space

For any smooth , multi-indices , we define its -Schwartz seminorm as

where and .
The Schwartz Space is then a space of rapidly decreasing functions:

e.g.

• In , the function is in for any multi-index ;
• Any compactly supported smooth function is in ;
  

Proposition

is a Fréchet space with the topology induced by the countable family of seminorms .

Proposition

For , the Schwartz space is dense in .

Distributions

Tempered Distribution

A tempered distribution is a continuous linear functional on . That is, the space of tempered distributions is the dual space of .

Instead of focusing on pointwise values, distributions generalize functions by how it acts on smooth “test functions” via a continuous linear functional. Here are some motivating examples:

e.g.

• Let be some integrable polynomially bounded function (i.e., there exists some polynomial such that for all ). Then it induces a tempered distribution by The integral converges because every decays faster at infinity than any polynomial.
• The familiar Dirac delta from physics is not a classical function but is perfectly defined as a distribution . It “picks out” the value of a test function at zero: More generally, the shifted delta at point works similarly: . In fact, many singular objects in physics (point sources, surface charges, impulsive forces) are rigorously modeled this way as distributions rather than classical functions.
  

Continuity Test Lemma

Distribution (Generalized Function)

A distribution or generalized function is a continuous (complex or real) linear functional on for some open set , where is the space of all smooth functions with compact support in , equipped with the usual LF-topology. The space of all distributions on is denoted by .

The ordinary locally integrable functions can be embedded into the space of distributions by the mapping where is the volume form on . In this way classical functions are identified with a subclass of distributions, often called regular distributions.

Smoothing Operator

A smoothing operator is a continuous linear map .

Schwartz Kernel Theorem for Scalar Distributions

Let be open. Then there is a one‑to‑one correspondence between continuous linear maps and distributions given by the formula for all and , where is defined by The distribution is called the Schwartz kernel of .

Distributional Sections of the Vector Bundle

We can further generalize the idea of distributions to smooth manifolds and vector bundles:

Distributional Section

Suppose is a smooth manifold and is a finite-rank vector bundle. Let be the space of compactly supported smooth sections of the dual bundle, then the space of distributional sections of is defined as the continuous dual

Schwartz Kernel Theorem for Distributional Sections

Let , be smooth manifolds, and let , be smooth finite‑rank vector bundles. Consider the space of compactly supported smooth sections with its LF‑topology and the space of distributional sections . Then every continuous linear operator is given by a unique distributional kernel such that for all and , we have where is defined by

The Principle Value

Let us first consider the improper integral . This is not well-defined because blows up at . But instead we can do the following: This is called the Cauchy principal value of the integral. It can be formulated as a distribution:

Cauchy Principle Value

For , we can define the distribution such that We shall prove that it is well-defined

Proposition

The principal value is the inverse distribution of the function and is almost the only distribution with this property: