Nuclear space

Under a nuclear space is understood in mathematics a special class of locally convex vector spaces. Many important applications in areas such as spaces of differentiable functions, are nuclear. While normed spaces, in particular Banach spaces and Hilbert spaces, generalizations but represent finite-dimensional vector spaces over (or ) while maintaining the standard with the loss of compactness properties, the focus is on the nuclear spaces which are not normalizable in the infinite-dimensional case, the compactness properties. Furthermore, unconditional convergence and absolute convergence of series in nuclear spaces prove to be equivalent. In this sense, the nuclear spaces are closer to the finite-dimensional spaces as the Banach spaces.

The going back to Alexander Grothendieck nuclear facilities can be introduced in many ways. As defined here, the easiest formulatable variant is selected, then follows a list of equivalent characterizations, which represent the same time a number of important properties of nuclear spaces. The following are examples and other properties.

• 3.1 Normed spaces
• 3.2 Quick falling consequences
• 3.3 Differentiable Functions
• 3.4 Test functions
• 3.5 Quick decreasing functions
• 3.6 Holomorphic functions

Definition

A locally convex space (always assumed to be Hausdorff space ) is called nuclear if every continuous linear operator is a nuclear operator for each Banach space.

Characterizations

Canonical pictures

Is a continuous seminorm on the locally convex space, then a closed subspace of and is a standard explained in the factor space. The completion of this normed space is denoted by. If a further continuous seminorm with so defines a continuous linear operator, which goes on steadily to a linear operator. The names of the local Banach spaces and the operators are called canonical pictures of.

These terms manages an internal characterization of nuclear spaces, that is, without reference to other rooms:

• A locally convex space is nuclear if and only if there is at any steady seminorm another continuous seminorm, so that the canonical map is a nuclear operator.

Of course, it is sufficient to restrict oneself to a directed system generating seminorms.

Hilbert spaces

Now following the characterizations back to the nuclear areas in the vicinity of the Hilbert space.

• A locally convex space is exact then nuclear, if there is a directed system of the topology generating seminorms, so that each local Banach space is a Hilbert space and a, gives to each, so that the canonical map is a Hilbert-Schmidt operator.

Is a Hermitian form with all ( that is, the Hermitian form is not negative), as defined by a half- norm on. Such semi-norms are called Hilbert semi-norms.

• A locally convex space is nuclear if and only if there are generating Hilbert seminorms a directed system, making it a, gives to each, so that the canonical map is a Hilbert-Schmidt operator.

Tensor

There are two important methods to equip the tensor product of two locally convex spaces with a suitable locally convex topology. Be and closed, absolutely convex environments. is the Minkowski functional of the absolutely convex hull of. Next denote the polar of the polar of analog and. This gives a more semi-norm on the definition.

The projective tensor or tensor is the Tensorproduktraum with the system of semi-norms, through which and the closed, absolutely convex neighborhoods of zero. Accordingly, the injective tensor or the tensor equipped with the system of semi-norms Tensorproduktraum.

Easily considered one that always applies, that is, is continuous. This figure is not a homeomorphism in general. The following applies:

• A locally convex space is nuclear if and only if is a homeomorphism for each locally convex space.

• A locally convex space is nuclear if and only if is a homeomorphism for each Banach space.

• A locally convex space is nuclear if and only if a homeomorphism.

This characterization is the original used by Grothendieck definition of nuclearity.

Bilinear forms

Is an absolutely convex neighborhood, the Polar is an absolutely absorbing and quantity in the vector space, is the corresponding Minkowski functional. A bilinear form is called nuclear if there is absolutely convex environments, as well as episodes in and with and for all and.

• A locally convex space is nuclear if and only if every continuous bilinear form for each locally convex space is nuclear.

• A locally convex space is nuclear if and only if every continuous bilinear form is nuclear for any Banach space.

This characterization of nuclear spaces is also called the abstract form of the sentence from the core.

Summability

Is an absolutely convex neighborhood, so is the associated Minkowski functional. is a neighborhood base of absolutely convex sets. Be provided with the semi-norms. The resulting locally convex space is called in an obvious way of absolute space Cauchy series. This definition does not require that the series converges.

Next we consider the space with the seminorms, where as above denotes the polar of and the base of neighborhoods by running. This space is called locally convex space of unconditional Cauchy series, because of the Riemannian or Steinitz rearrangement theorem follows easily that also with each permuted sequence is within.

Both and are independent of the particular choice of the base of neighborhoods. The nuclear spaces is now proving to be those in which coincide absolute Cauchy series and unconditional Cauchy series:

• A locally convex space is nuclear if and only if as sets and as topological spaces.

Set of Komura Komura -

The presented here goes back to T. Komura and Y. Komura theorem shows that the result stated in the examples space of rapidly falling Following is a generator of all nuclear facilities.

• A locally convex space is nuclear if and only if there are a lot, so that is isomorphic to a subspace of.

Examples

Normed spaces

Among the normed spaces are precisely the finite- nuclear.

Quick falling consequences

Be the semi-norms. This space is called locally convex space of rapidly falling consequences and is according to the above set of Komura Komura - a prototype of a nuclear space.

Differentiable functions

Important examples are also spaces of differentiable functions. Be open and is the space of infinitely differentiable functions with the semi-norms, where and compact. It was used for the multi- index notation. Then a nuclear space.

Test functions

Be open and the subspace of infinitely differentiable functions with compact support in. For compact is the space of functions with support in K with the induced subspace topology. Then there is a finest locally convex topology that makes all embeddings steadily. with this topology is the area of the test functions, and plays an important role in the distribution theory. is an example of a non- metrizable nuclear space.

Quick decreasing functions

Be the space of all functions for which all suprema are finite. It was taken again by the multi- index notation use. The space with the seminorms is called space of functions quickly falling and is also nuclear.

Holomorphic functions

Be open and the space of all holomorphic functions. Then, with the semi-norms, which is compact, a nuclear space.

Permanenzeigenschaften

Nuclear rooms have very good Permanenzeigenschaften. Subspaces factor spaces by closed subspaces, any products, countable direct sums, tensor products and nuclear spaces are completions nuclear again.

Properties

• Nuclear spaces have the approximation property.

• In metrizable, nuclear generalization of the Steinitz Umordnungssatzes, is carried out as in the article about the reordering of rows applies.

• Complete nuclear spaces are Schwartz spaces.

• Nuclear Fréchet spaces are Montel spaces.

• The strong dual space of a nuclear space is a barreled space.

• In quasi- complete nuclear spaces of the theorem of Bolzano - Weierstrass holds, ie a lot is compact if it is closed and bounded.

• Quasi Complete nuclear spaces are reflexive. Therefore, virtually complete, quasi- barreled, nuclear spaces are reflexive.