Nuclear space

In mathematics, a nuclear space is a special class of locally convex vector spaces . Many rooms that are important in the applications, e.g. B. Spaces of differentiable functions are nuclear. While normalized spaces , in particular Banach spaces or Hilbert spaces , represent generalizations of finite-dimensional vector spaces over ( or ) maintaining the norm but losing compactness properties , the focus of nuclear spaces, which cannot be normalized in the infinite-dimensional case , is on 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 than the Banach spaces. ${\displaystyle {\mathbb {K} }}$ ${\displaystyle {\mathbb {R} }}$ ${\displaystyle {\mathbb {C} }}$

The nuclear spaces that go back to Alexander Grothendieck can be introduced in a variety of ways. The most easily formulated variant is chosen as the definition, followed by a list of equivalent characterizations that simultaneously represent a number of important properties of nuclear spaces. Examples and other properties follow.

definition

A locally convex space (always assumed to be Hausdorff space ) is called nuclear if for every Banach space every continuous linear operator is a nuclear operator . ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle E\rightarrow F}$

Characterizations

Canonical illustrations

If there is a continuous semi-norm on the locally convex space , then a closed sub-vector space is explained by and through a norm on the factor space . The completion of this standardized space is denoted by. Is a more steady with semi-norm , is defined a steady linear operator , the continuously at a linear operator continue leaves. The names of the local Banach spaces and the operators are called canonical figures of . ${\displaystyle p}$ ${\displaystyle E}$ ${\displaystyle N_{p}:=\{x\in E;p(x)=0\}}$ ${\displaystyle E}$ ${\displaystyle \|x+N_{p}\|_{p}:=p(x)}$ ${\displaystyle E_{p}:=E/N_{p}}$ ${\displaystyle B_{p}}$ ${\displaystyle q}$ ${\displaystyle p\leq q}$ ${\displaystyle x+N_{q}\mapsto x+N_{p}}$ ${\displaystyle E_{q}\rightarrow E_{p}}$ ${\displaystyle \kappa _{qp}\colon B_{q}\rightarrow B_{p}}$ ${\displaystyle B_{p}}$ ${\displaystyle \kappa _{qp}}$ ${\displaystyle E}$

With these terms, an internal characterization of nuclear spaces is possible, i.e. without reference to other spaces:

• A locally convex space is nuclear if and only if there is a further continuous semi-norm for every continuous semi- norm , so that the canonical mapping is a nuclear operator . ${\displaystyle p}$ ${\displaystyle q\geq p}$ ${\displaystyle \kappa _{qp}}$

It is of course sufficient to restrict oneself to a directed system of generating semi-norms.

Hilbert dreams

The characterizations that now follow move the nuclear spaces closer to the Hilbert spaces.

• A locally convex space is exactly then nuclear , if there is a directional system are of the topology generating seminorms so that each local Banach space , a Hilbert space , and, for every one , is such that the canonical map a Hilbert-Schmidt operator is. ${\displaystyle {\mathcal {P}}}$ ${\displaystyle B_{p},p\in {\mathcal {P}}}$ ${\displaystyle p\in {\mathcal {P}}}$ ${\displaystyle q\in {\mathcal {P}},q\geq p}$ ${\displaystyle \kappa _{qp}}$

Is a Hermitian form on with for all (i. E. The Hermitian form is non-negative), then by a semi-norm on defined. Such semi-norms are called Hilbert semi-norms . ${\displaystyle \langle \cdot ,\cdot \rangle }$ ${\displaystyle E}$ ${\displaystyle \langle x,x\rangle \geq 0}$ ${\displaystyle x\in E}$ ${\displaystyle x\mapsto \langle x,x\rangle ^{\frac {1}{2}}}$ ${\displaystyle E}$

• A locally convex space is nuclear if and only if there is a directed system of generating Hilbert semi-norms such that there is for each one , so that the canonical mapping is a Hilbert-Schmidt operator. ${\displaystyle {\mathcal {P}}}$ ${\displaystyle p\in {\mathcal {P}}}$ ${\displaystyle q\in {\mathcal {P}},q\geq p}$ ${\displaystyle \kappa _{qp}}$

Tensor products

There are two important methods of equipping the tensor product of two locally convex spaces with a suitable locally convex topology. Let and be closed, absolutely convex zero neighborhoods. be the Minkowski functional of the absolutely convex hull of . Further denote the polar of and analogously the polar of . Another semi-norm is obtained through the definition . ${\displaystyle E\otimes F}$ ${\displaystyle U\subset E}$ ${\displaystyle V\subset F}$ ${\displaystyle \pi _{U,V}}$ ${\displaystyle U\otimes V:=\{x\otimes y;x\in U,y\in V\}}$ ${\displaystyle U^{\circ }:=\{\varphi \in E^{\,\prime }\,;\,{\rm {{Re}(\varphi (x))\leq 1\ \forall \,x\in U\}}}}$ ${\displaystyle U}$ ${\displaystyle V^{\circ }}$ ${\displaystyle V}$ ${\displaystyle \epsilon _{U,V}}$ ${\displaystyle E\otimes F}$ ${\displaystyle \textstyle \epsilon _{U,V}(\sum _{j=1}^{n}x_{j}\otimes y_{j}):=\sup\{|\sum _{j=1}^{n}\phi (x_{j})\psi (y_{j})|;\,\phi \in U^{\circ },\psi \in V^{\circ }\}}$

The projective tensor or -Tensorprodukt is the Tensorproduktraum with the system of seminorms , wherein and through the closed, absolutely convex zero environments. Accordingly, the injective tensor or -Tensorprodukt the with the system of seminorms equipped Tensorproduktraum. ${\displaystyle \pi }$ ${\displaystyle E\otimes _{\pi }F}$ ${\displaystyle \pi _{U,V}}$ ${\displaystyle U\subset E}$ ${\displaystyle V\subset F}$ ${\displaystyle \epsilon }$ ${\displaystyle E\otimes _{\epsilon }F}$ ${\displaystyle \epsilon _{U,V}}$

It is easy to consider that it always holds, i. H. is steady. This mapping is generally not a homeomorphism . The following applies: ${\displaystyle \epsilon _{U,V}\leq \pi _{U,V}}$ ${\displaystyle id:E\otimes _{\pi }F\rightarrow E\otimes _{\epsilon }F}$

• A locally convex space is nuclear if and only if there is a homeomorphism for every locally convex space . ${\displaystyle E}$ ${\displaystyle id:E\otimes _{\pi }F\rightarrow E\otimes _{\epsilon }F}$ ${\displaystyle F}$

• A locally convex space is nuclear if and only if there is a homeomorphism for every Banach space . ${\displaystyle E}$ ${\displaystyle id:E\otimes _{\pi }F\rightarrow E\otimes _{\epsilon }F}$ ${\displaystyle F}$

• A locally convex space is nuclear if and only if there is a homeomorphism. ${\displaystyle E}$ ${\displaystyle id:E\otimes _{\pi }\ell ^{1}\rightarrow E\otimes _{\epsilon }\ell ^{1}}$

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

Bilinear forms

If there is an absolutely convex zero neighborhood, then the polar is an absolutely convex and absorbing set in vector space , be the associated Minkowski functional. A bilinear form is called nuclear if there are absolutely convex zero neighborhoods and as well as sequences in and in with and for all and . ${\displaystyle U\subset E}$ ${\displaystyle U^{\circ }}$ ${\displaystyle \textstyle E_{U^{\circ }}:=\bigcup _{\lambda >0}\lambda U^{\circ }\subset E\,'}$ ${\displaystyle \|\cdot \|_{U^{\circ }}}$ ${\displaystyle b:E\times F\rightarrow {\mathbb {K} }}$ ${\displaystyle U\subset E}$ ${\displaystyle V\subset F}$ ${\displaystyle (a_{n})_{n}}$ ${\displaystyle E_{U^{\circ }}}$ ${\displaystyle (b_{n})_{n}}$ ${\displaystyle E_{V^{\circ }}}$ ${\displaystyle \textstyle \sum _{n=1}^{\infty }\|a_{n}\|_{U^{\circ }}\|b_{n}\|_{V^{\circ }}<\infty }$ ${\displaystyle \textstyle b(x,y)=\sum _{n=1}^{\infty }a_{n}(x)b_{n}(y)}$ ${\displaystyle x\in E}$ ${\displaystyle y\in F}$

• A locally convex space is nuclear if and only if every continuous bilinear form for every locally convex space is nuclear. ${\displaystyle E\times F\rightarrow {\mathbb {K} }}$ ${\displaystyle F}$

• A locally convex space is nuclear if and only if every continuous bilinear form is nuclear for every Banach space . ${\displaystyle E\times F\rightarrow {\mathbb {K} }}$ ${\displaystyle F}$

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

Summability

If there is an absolutely convex zero neighborhood, let the associated Minkowski functional be. be a zero neighborhood basis from absolutely convex sets. Be provided with the semi-norms . The resulting locally convex space is obviously called the space of the absolute Cauchy series. This definition does not require the series to converge in. ${\displaystyle U\subset E}$ ${\displaystyle p_{U}}$ ${\displaystyle {\mathcal {U}}}$ ${\displaystyle \textstyle \ell ^{1}\{E\}:=\{(x_{n})_{n}\in E^{\mathbb {N} }\,;\,\sum _{n=1}^{\infty }p_{U}(x_{n})<\infty \,\forall U\in {\mathcal {U}}\}}$ ${\displaystyle \textstyle q_{U}((x_{n})_{n}):=\sum _{n=1}^{\infty }p_{U}(x_{n})}$ ${\displaystyle \textstyle \sum _{n=1}^{\infty }x_{n}}$ ${\displaystyle E}$

Further we consider the space with the semi-norms , where as above the polar of is denoted and the zero neighborhood base passes through. This locally convex space is called space of unconditional Cauchy series, because of the Riemann or steinitzschen rearrangement theorem easily follows that with , any permuted result in lies. ${\displaystyle \ell ^{1}[E]:=\{(x_{n})_{n}\in E^{\mathbb {N} }\,;\,(f(x_{n}))_{n}\in \ell ^{1}\,\,\forall f\in E'\}}$ ${\displaystyle \textstyle \epsilon _{U}((x_{n})_{n}):=\sup _{f\in U^{\circ }}\sum _{n=01}^{\infty }|f(x_{n})|}$ ${\displaystyle U^{\circ }}$ ${\displaystyle U}$ ${\displaystyle U}$ ${\displaystyle {\mathcal {U}}}$ ${\displaystyle (x_{n})_{n}}$ ${\displaystyle (x_{\sigma (n)})_{n}}$ ${\displaystyle \ell ^{1}[E]}$

Both and are independent of the particular choice of zero neighborhood base. The nuclear spaces now turn out to be those in which absolute Cauchy series and unconditional Cauchy series coincide: ${\displaystyle \ell ^{1}\{E\}}$ ${\displaystyle \ell ^{1}[E]}$

• A locally convex space is nuclear if and only if as sets and as topological spaces. ${\displaystyle \ell ^{1}\{E\}=\ell ^{1}[E]}$

Theorem of Kōmura-Kōmura

The sentence presented here, which goes back to T. Kōmura and Y. Kōmura, shows that the sequence space of the rapidly falling sequences given in the examples is a generator of all nuclear spaces. ${\displaystyle s}$

• A locally convex space is nuclear if and only if there is a set such that is isomorphic to a subspace of . ${\displaystyle E}$ ${\displaystyle I}$ ${\displaystyle E}$ ${\displaystyle s^{I}}$

Examples

Standardized spaces

Among the normalized spaces , it is precisely the finite-dimensional spaces that are nuclear.

Rapidly falling episodes

Be with the semi-norms . This locally convex space is called the space of rapidly falling sequences and, according to the above sentence by Kōmura-Kōmura, is a prototype of a nuclear space. ${\displaystyle s:=\{(x_{n})_{n}\in {\mathbb {K} }^{\mathbb {N} };(x_{n}n^{k})_{n}\in \ell ^{1}\,\,\forall k\in {\mathbb {N} }\}}$ ${\displaystyle \textstyle p_{k}((x_{n})_{n}):=\sum _{n=1}^{\infty }|x_{n}n^{k}|}$

Differentiable functions

Important examples are also spaces with differentiable functions. Be open and the space of any number of differentiable functions with the semi-norms , where and is compact. It was for the multiindex used notation. Then there is a nuclear space. ${\displaystyle \Omega \subset {\mathbb {R} }^{n}}$ ${\displaystyle {\mathcal {E}}(\Omega )}$ ${\displaystyle f:\Omega \rightarrow {\mathbb {R} }}$ ${\displaystyle p_{K,m}(f):=\sup _{|\alpha |\leq m}\sup _{x\in K}|D^{\alpha }f(x)|}$ ${\displaystyle m\in {\mathbb {N} }}$ ${\displaystyle K\subset \Omega }$ ${\displaystyle \alpha =(\alpha _{1},\ldots ,\alpha _{n})}$ ${\displaystyle {\mathcal {E}}(\Omega )}$

Test functions

Be open and the subspace of any number of differentiable functions with a compact carrier in . For compact, let the space of the functions with carrier in K with the subspace topology induced by . Then there is a finest locally convex topology that makes all embeddings continuous. with this topology the space is called the test functions and plays an important role in distribution theory . is an example of a non- metrizable nuclear space. ${\displaystyle \Omega \subset {\mathbb {R} }^{n}}$ ${\displaystyle {\mathcal {D}}(\Omega )\subset {\mathcal {E}}(\Omega )}$ ${\displaystyle \Omega }$ ${\displaystyle K\subset \Omega }$ ${\displaystyle {\mathcal {D}}_{K}(\Omega )}$ ${\displaystyle {\mathcal {E}}(\Omega )}$ ${\displaystyle {\mathcal {D}}(\Omega )}$ ${\displaystyle {\mathcal {D}}_{K}(\Omega )\subset {\mathcal {D}}(\Omega )}$ ${\displaystyle {\mathcal {D}}(\Omega )}$ ${\displaystyle {\mathcal {D}}(\Omega )}$

Fast falling functions

Be the space of all functions for which all suprema are finite. The multi-index notation was used again. The space with the semi-norms is called the space of rapidly falling functions and is also nuclear. ${\displaystyle {\mathcal {S}}({\mathbb {R} }^{n})}$ ${\displaystyle f:{\mathbb {R} }^{n}\rightarrow {\mathbb {R} }}$ ${\displaystyle \textstyle p_{k,m}(f):=\sup _{|\alpha |\leq k}\sup _{x\in {\mathbb {R} }^{n}}|(1+|x|^{2})^{m}D^{\alpha }f(x)|}$ ${\displaystyle {\mathcal {S}}({\mathbb {R} }^{n})}$ ${\displaystyle \{p_{k,m};\,k,m\in {\mathbb {N} }_{0}\}}$

Holomorphic functions

Be open and the space of all holomorphic functions . Then with the semi-norms , where is compact, there is a nuclear space. ${\displaystyle \Omega \subset {\mathbb {C} }}$ ${\displaystyle {\mathcal {H}}(\Omega )}$ ${\displaystyle \Omega \rightarrow {\mathbb {C} }}$ ${\displaystyle {\mathcal {H}}(\Omega )}$ ${\displaystyle p_{K}:=p_{K,0}}$ ${\displaystyle K\subset \Omega }$

Permanent properties

Nuclear spaces have very good permanence properties. Subspaces, factor spaces after closed subspaces, arbitrary products , countable direct sums , tensor products and completions of nuclear spaces are again nuclear.

properties

• Nuclear spaces have the approximation property .

• In metrizable, nuclear ones, the generalization of Steinitz's rearrangement theorem applies , as explained in the article on the rearrangement of series .

• Complete nuclear spaces are Schwartz spaces .

• Nuclear Fréchet rooms are Montel rooms .

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

• In quasi-complete nuclear spaces, the Bolzano-Weierstrass theorem applies , i.e. H. a set is compact if and only if it is closed and bounded .

• Quasi-complete nuclear spaces are semi- reflective . Hence, quasi-complete, quasi-tiled , nuclear spaces are reflexive .

literature

• A Grothendieck: Résumé des résultats essentiels dans la théorie des produits tensoriels topologiques et des espaces nucléaires , Ann. Inst. Fourier, 4: 73-112 (1954)
• A. Pietsch, Nuclear locally convex spaces , Akademie-Verlag, Berlin, 1969
• K. Floret, J. Wloka: Introduction to the Theory of Locally Convex Spaces , Lecture Notes in Mathematics 56, 1968
• HH Schaefer: Topological Vector Spaces , Springer, 1971 ISBN 0-387-98726-6
• H. Jarchow: Locally Convex Spaces , Teubner, Stuttgart 1981 ISBN 3-519-02224-9
• R. Meise, D. Vogt: Introduction to Functional Analysis , Vieweg, 1992 ISBN 3-528-07262-8