Injective tensor product

The injective tensor product is an extension of the tensor products of vector spaces considered in mathematics for the case that additional topologies are present on the vector spaces. In this situation it makes sense to also want to explain a topology on the tensor product of the spaces. Among the many ways of doing this, the projective tensor product and the injective tensor product to be discussed here are natural choices.

First, the more accessible case of normalized spaces and Banach spaces is discussed, then the generalizations in the theory of locally convex spaces are discussed.

The construction for standardized spaces and Banach spaces goes back to Robert Schatten , the generalizations to locally convex spaces were achieved by Alexander Grothendieck .

Standardized spaces

The tensor product of two normalized spaces can also be made into a normalized space as follows.

construction

Be and standardized spaces. Each two continuous, linear functionals and define a bilinear mapping . According to the universal definition of the tensor product, this induces a linear mapping , which is usually referred to as. You now bet for ${\displaystyle (E,\|\cdot \|_{1})}$ ${\displaystyle (F,\|\cdot \|_{2})}$ ${\displaystyle f\in E\,'}$ ${\displaystyle g\in F\,'}$ ${\displaystyle E\times F\rightarrow {\mathbb {K} },\,(x,y)\mapsto f(x)g(y)}$ ${\displaystyle E\otimes F\rightarrow {\mathbb {K} },\,\sum _{i=1}^{n}x_{i}\otimes y_{i}\mapsto \sum _{i=1}^{n}f(x_{i})g(y_{i})}$ ${\displaystyle f\otimes g}$ ${\displaystyle z\in E\otimes F}$

${\displaystyle \|z\|_{\epsilon }:=\sup\{|(f\otimes g)(z)|;\,f\in E\,',g\in F\,',\|f\|_{1}\leq 1,\|g\|_{2}\leq 1\}}$,

where the norms on the dual spaces are designated as in the output spaces . This definition gives a norm on the tensor product, the so-called injective tensor product of the norms and . If one provides this norm, one calls the injective tensor product or also the - tensor product of the normed spaces and and writes for it . The injective tensor product is also called the weak tensor product . ${\displaystyle \|\cdot \|_{1}}$ ${\displaystyle \|\cdot \|_{2}}$ ${\displaystyle E\otimes F}$ ${\displaystyle E\otimes F}$ ${\displaystyle \epsilon }$ ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle E\otimes _{\epsilon }F}$

properties

Are in the situation of the above definition , then applies . ${\displaystyle x\in E,y\in F}$ ${\displaystyle \|x\otimes y\|_{\epsilon }=\|x\|_{1}\cdot \|y\|_{2}}$

It always holds , where denotes the projective tensor product . ${\displaystyle \|z\|_{\epsilon }\leq \|z\|_{\pi }}$ ${\displaystyle \|\cdot \|_{\pi }}$

Each defines a continuous linear operator by putting. It is easy to show that the norm of matches the operator norm of . This could have been used as an alternative definition for the norm, but the symmetry with which and go into the definition would not have been as obvious as in the definition given above. ${\displaystyle z=\sum _{i=1}^{n}x_{i}\otimes y_{i}}$ ${\displaystyle T_{z}:E\,'\rightarrow F}$ ${\displaystyle T_{z}(f):=\sum _{i=1}^{n}f(x_{i})y_{i}}$ ${\displaystyle \epsilon }$ ${\displaystyle z}$ ${\displaystyle T_{z}}$ ${\displaystyle \epsilon }$ ${\displaystyle E}$ ${\displaystyle F}$

Banach rooms

The injective tensor product of two Banach spaces and is usually not complete, so that the formation of the tensor product leads out of the category of Banach spaces. In order to stay in the category of Banach spaces, one has to complete. ${\displaystyle (E,\|\cdot \|_{1})}$ ${\displaystyle (F,\|\cdot \|_{2})}$

One therefore defines as the completion of the normalized space and calls the injective tensor product in the category of Banach spaces. ${\displaystyle E{\hat {\otimes }}_{\epsilon }F}$ ${\displaystyle E\otimes _{\epsilon }F}$ ${\displaystyle E{\hat {\otimes }}_{\epsilon }F}$

Hilbert dreams

If a Hilbert space is , then according to the above there is an isometric embedding in the space of the continuous linear operators . One can show that with this identification the tensor product coincides exactly with the compact operators , that is, it holds . In particular, this example shows that the injective tensor product of Hilbert spaces is generally not a Hilbert space. ${\displaystyle H}$ ${\displaystyle H\,'\otimes H\subset L(H)}$ ${\displaystyle H}$ ${\displaystyle H\,'{\hat {\otimes }}_{\epsilon }H\cong K(H)}$

The tensor product with spaces of continuous functions

If a space is compact , then denote the Banach space of continuous functions with the supremum norm . be another Banach space and be the Banach space of the -valent continuous functions with the supremum norm. Then there is an isometric embedding with a dense image, that is, this embedding continues to an isometric isomorphism between and . That is written briefly and concisely as ${\displaystyle X}$ ${\displaystyle C(X)}$ ${\displaystyle X\rightarrow {\mathbb {K} }}$ ${\displaystyle E}$ ${\displaystyle C(X,E)}$ ${\displaystyle E}$ ${\displaystyle C(X)\otimes _{\epsilon }E\rightarrow C(X,E),\sum _{i=1}^{n}f_{i}\otimes y_{i}\mapsto \sum _{i=1}^{n}f(\cdot )\otimes y_{i}}$ ${\displaystyle C(X){\hat {\otimes }}_{\epsilon }E}$ ${\displaystyle C(X,E)}$

${\displaystyle C(X){\hat {\otimes }}_{\epsilon }E\cong C(X,E)}$.

In particular, for two compact spaces and the expected isometrics are obtained ${\displaystyle X_{1}}$ ${\displaystyle X_{2}}$

${\displaystyle C(X_{1}){\hat {\otimes }}_{\epsilon }C(X_{2})\,\cong \,C(X_{1},C(X_{2}))\,\cong \,C(X_{1}\times X_{2})}$.

Tensor product with ℓ ¹

Let it be the sequence space of the absolutely convergent, real series and a Banach space. As is known, the projective tensor product can be identified with the space of the absolutely convergent series in . A similar characterization succeeds for the injective tensor product if one replaces the absolute convergence with unconditional convergence . ${\displaystyle \ell ^{1}}$ ${\displaystyle E}$ ${\displaystyle \ell ^{1}{\hat {\otimes }}_{\pi }E}$ ${\displaystyle \ell ^{1}(E)}$ ${\displaystyle E}$ ${\displaystyle \ell ^{1}{\hat {\otimes }}_{\epsilon }E}$

Let it be the space of the unconditionally convergent series in . If there is such a series, then it is absolutely convergent for each . It is even true that ${\displaystyle \ell ^{1}[E]}$ ${\displaystyle E}$ ${\displaystyle (x_{n})_{n}}$ ${\displaystyle \sum _{n=1}^{\infty }f(x_{n})}$ ${\displaystyle f\in E'}$

${\displaystyle \|(x_{n})_{n}\|:=\sup \left\{\sum _{n=1}^{\infty }|f(x_{n})|;\,f\in E',\|f\|\leq 1\right\}}$

is finite and defines a norm which makes it a Banach space. Then one can show that the bilinear mapping ${\displaystyle \ell ^{1}[E]}$ ${\displaystyle \ell ^{1}[E]}$

${\displaystyle \ell ^{1}\times E\rightarrow \ell ^{1}[E],((a_{n})_{n},x)\mapsto (a_{n}x)_{n}}$

an isometric illustration

${\displaystyle \ell ^{1}\otimes _{\epsilon }E\rightarrow \ell ^{1}[E]}$

which induces an isometric isomorphism

${\displaystyle \ell ^{1}{\hat {\otimes }}_{\epsilon }E\rightarrow \ell ^{1}[E]}$

continues.

Local convex spaces

The construction of the injective tensor product can be generalized to the case of locally convex spaces as follows.

definition

Let and be two locally convex spaces, and let and be absolutely convex zero neighborhoods. Further denote the polar of and analogously the polar of . A semi-norm is obtained through the definition . ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle U\subset E}$ ${\displaystyle V\subset F}$ ${\displaystyle U^{\circ }:=\{\varphi \in E^{\,\prime }\,;\,{\rm {{Re}(\varphi (x))\leq 1\ \forall \,x\in U\}}}}$ ${\displaystyle U}$ ${\displaystyle V^{\circ }}$ ${\displaystyle V}$ ${\displaystyle \epsilon _{U,V}}$ ${\displaystyle E\otimes F}$ ${\displaystyle \epsilon _{U,V}(\sum _{j=1}^{n}x_{j}\otimes y_{j}):=\sup\{|\sum _{j=1}^{n}f(x_{j})g(y_{j})|;\,f\in U^{\circ },g\in V^{\circ }\}}$

The injective tensor or -Tensorprodukt is with the system of seminorms equipped Tensorproduktraum, wherein and the absolutely convex zero environments and run through. This generalizes the definition of the injective tensor product of normalized spaces. ${\displaystyle \epsilon }$ ${\displaystyle E\otimes _{\epsilon }F}$ ${\displaystyle \epsilon _{U,V}}$ ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle E}$ ${\displaystyle F}$

As in the case of standardized spaces, the completion of is denoted by. ${\displaystyle E\otimes _{\epsilon }F}$ ${\displaystyle E{\hat {\otimes }}_{\epsilon }F}$

Stability properties

Some classes of locally convex spaces are stable towards the formation of the injective tensor product. Include and both to one of the classes ${\displaystyle E}$ ${\displaystyle F}$

• standardized spaces
• Metrizable locally convex spaces
• nuclear spaces
• Schwartz rooms ,

so also belongs to this class. ${\displaystyle E\otimes _{\epsilon }F}$

• Are and Fréchet - Montel rooms, so too . ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle E{\hat {\otimes }}_{\epsilon }F}$

The tensor product with spaces of continuous functions

Let it be a completely regular space and denote the space of continuous functions with the topology of uniform convergence on compact sets. If there is another locally convex space, then let it be the space of -valent continuous functions with the topology of uniform convergence on compact sets. Then there is natural isomorphism ${\displaystyle X}$ ${\displaystyle C(X)_{c}}$ ${\displaystyle X\rightarrow {\mathbb {K} }}$ ${\displaystyle E}$ ${\displaystyle C(X,E)_{c}}$ ${\displaystyle E}$

${\displaystyle C(X)_{c}{\hat {\otimes }}_{\epsilon }E\cong C(X,E)_{c}}$,

if complete and is a kelley space . A Kelley space is called if a function is continuous if its constraints are continuous on compact subsets. This is the case , for example, with locally compact spaces . ${\displaystyle E}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle X\rightarrow {\mathbb {K} }}$

See also

• Hilbert space tensor product
• Pettis integral
• Vectorial measure

Individual evidence

1. ^ R. Schatten: A theory of cross spaces. Annals of Mathematical Studies 26, Princeton, NJ (1950)
2. ^ A. Grothendieck: Produits tensoriels topologiques et espaces nucléaires. Mem. Amer. Math. Soc. 16 (1955)
3. ^ Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces , Springer-Verlag 2002, ISBN 1-85233-437-1 , Chapter 3: The Injective Tensor Product
4. ↑ AY Helemskii: The homology of Banach and Topological Algebras. Kluwer Academic Publishers (1989), ISBN 0-7923-0217-6 , Chapter II, Definition 2.55
5. ^ Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces , Springer-Verlag 2002, ISBN 1-85233-437-1 , chapter 3.2
6. ^ Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces , Springer-Verlag 2002, ISBN 1-85233-437-1 , example 3.4
7. ^ H. Jarchow: Locally Convex Spaces. Teubner, Stuttgart (1981), ISBN 3-519-02224-9