Projective tensor product

The projective tensor product is an extension of the tensor products of vector spaces considered in mathematics to 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 injective tensor product and the projective tensor product to be discussed here are natural choices.

The investigation of the projective tensor product of locally convex spaces goes back to Alexander Grothendieck . Some results about Banach spaces were previously obtained by Robert Schatten .

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.

Standardized spaces

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

definition

Be and standardized spaces. The elements of the tensor product can be written in the form , whereby this sum representation is not unambiguous. One defines ${\displaystyle (E,\|\cdot \|_{1})}$ ${\displaystyle (F,\|\cdot \|_{2})}$ ${\displaystyle z\in E\otimes F}$ ${\displaystyle z=\sum _{i=1}^{n}x_{i}\otimes y_{i}}$

${\displaystyle \|z\|_{\pi }:=\inf\{\sum _{i=1}^{n}\|x_{i}\|_{1}\cdot \|y_{i}\|_{2};\,n\in {\mathbb {N} },x_{i}\in E,y_{i}\in F,z=\sum _{i=1}^{n}x_{i}\otimes y_{i}\}}$,

so we get a norm on the tensor product . This norm is called the projective tensor product of norms and . If one provides this norm, one calls the projective tensor product or also the - tensor product of the normed spaces and and writes for it . ${\displaystyle E\otimes F}$ ${\displaystyle \|\cdot \|_{1}}$ ${\displaystyle \|\cdot \|_{2}}$ ${\displaystyle E\otimes F}$ ${\displaystyle E\otimes F}$ ${\displaystyle \pi }$ ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle E\otimes _{\pi }F}$

properties

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

If there is a continuous, bilinear mapping between normalized spaces, then this induces a uniquely determined continuous, linear mapping , whereby for all . The following applies to the operator norm . ${\displaystyle B:E\times F\rightarrow G}$ ${\displaystyle B_{0}:E\otimes F\rightarrow G}$ ${\displaystyle B(x,y)=B_{0}(x\otimes y)}$ ${\displaystyle x\in E,y\in F}$ ${\displaystyle \|B_{0}\|=\sup\{\|B(x,y)\|;\,x\in E,\|x\|_{1}\leq 1,y\in F,\|y\|_{2}\leq 1\}}$

Therefore the tensor product is in the category of normalized spaces with the continuous linear mappings as morphisms in the sense of the universal definition of the tensor product . ${\displaystyle \otimes _{\pi }}$

Banach rooms

The projective 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. ${\displaystyle (E,\|\cdot \|_{1})}$ ${\displaystyle (F,\|\cdot \|_{2})}$

definition

One defines as the completion of the normalized space and calls the projective tensor product in the category of Banach spaces. This definition is particularly motivated by the following universal property. ${\displaystyle E{\hat {\otimes }}_{\pi }F}$ ${\displaystyle E\otimes _{\pi }F}$ ${\displaystyle E{\hat {\otimes }}_{\pi }F}$

Universal property

If there is a continuous, bilinear mapping between Banach spaces, then there is exactly one continuous, linear mapping with for all . The same applies to the operator norm as in the case of normalized spaces . ${\displaystyle B:E\times F\rightarrow G}$ ${\displaystyle B_{0}:E{\hat {\otimes }}_{\pi }F\rightarrow G}$ ${\displaystyle B(x,y)=B_{0}(x\otimes y)}$ ${\displaystyle x\in E,y\in F}$ ${\displaystyle \|B_{0}\|=\sup\{\|B(x,y)\|;\,x\in E,\|x\|_{1}\leq 1,y\in F,\|y\|_{2}\leq 1\}}$

So the tensor product is in the category of Banach spaces with the continuous linear mappings as morphisms in the sense of the universal definition of the tensor product. ${\displaystyle {\hat {\otimes }}_{\pi }}$

Representation of the elements

Each element has a representation with , such view as absolutely convergent series is not unique. The formula applies ${\displaystyle z\in E{\hat {\otimes }}_{\pi }F}$ ${\displaystyle z=\sum _{i=1}^{\infty }x_{i}\otimes y_{i}}$ ${\displaystyle \sum _{i=1}^{\infty }\|x_{i}\|_{1}\cdot \|y_{i}\|_{2}<\infty }$

${\displaystyle \|z\|_{\pi }:=\inf\{\sum _{i=1}^{\infty }\|x_{i}\|_{1}\cdot \|y_{i}\|_{2};\,x_{i}\in E,y_{i}\in F,z=\sum _{i=1}^{\infty }x_{i}\otimes y_{i},\sum _{i=1}^{\infty }\|x_{i}\|_{1}\cdot \|y_{i}\|_{2}<\infty \}}$.

Dual spaces

The dual space of a projective tensor product can be identified with the space of continuous, linear operators from into the dual space from . If such an operator is, then is ${\displaystyle E{\hat {\otimes }}_{\pi }F}$ ${\displaystyle L(E,F')}$ ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle S:E\rightarrow F'}$

${\displaystyle \psi _{S}:E\otimes F\rightarrow \mathbb {R} ,\,\sum _{i=1}^{n}x_{i}\otimes y_{i}\mapsto \sum _{i=1}^{n}(S(x_{i}))(y_{i})}$

a -stetiges linear functional whose standard with the operator norm match, it can be equal to a standard continuous linear functional thus to continue. Then you can show that ${\displaystyle \|\cdot \|_{\pi }}$ ${\displaystyle {\overline {\psi _{S}}}}$ ${\displaystyle E{\hat {\otimes }}_{\pi }F}$ ${\displaystyle \psi }$

${\displaystyle \psi :L(E,F')\rightarrow (E{\hat {\otimes }}_{\pi }F)',\,S\mapsto {\overline {\psi _{S}}}}$

is an isometric isomorphism. This is the sense in which identification is to be understood. ${\displaystyle L(E,F')\cong (E{\hat {\otimes }}_{\pi }F)'}$

The tensor product with L ¹ spaces

Let there be a measure space and a Banach space. Be the Banach space of all equivalence classes of measurable functions with which two measurable functions are equivalent if they -almost everywhere the same, that is, when most within one - zero quantity take on different values. According to the universal property, the bilinear mapping induces a continuous linear mapping . The theorem now applies that this mapping is an isometric isomorphism . That is written briefly and concisely as ${\displaystyle (X,\Sigma ,\mu )}$ ${\displaystyle (E,\|\cdot \|)}$ ${\displaystyle L^{1}(X,\Sigma ,\mu ,E)}$ ${\displaystyle f:X\rightarrow E}$ ${\displaystyle \|f\|_{1}:=\int _{X}\|f\|{\rm {d}}\mu (t)<\infty }$ ${\displaystyle \mu }$ ${\displaystyle \mu }$ ${\displaystyle L^{1}(X,\Sigma ,\mu )\times E\rightarrow L^{1}(X,\Sigma ,\mu ,E),(f,x)\mapsto f(\cdot )x}$ ${\displaystyle L^{1}(X,\Sigma ,\mu ){\hat {\otimes }}_{\pi }E\rightarrow L^{1}(X,\Sigma ,\mu ,E)}$

${\displaystyle L^{1}(X,\Sigma ,\mu ){\hat {\otimes }}_{\pi }E\cong L^{1}(X,\Sigma ,\mu ,E)}$.

Banach algebras

Be and Banach algebras . Then the definition continues to a multiplication on which makes a Banach algebra, that is, the norm is submultiplicative . ${\displaystyle (E,\|\cdot \|_{1})}$ ${\displaystyle (F,\|\cdot \|_{2})}$ ${\displaystyle (x_{1}\otimes y_{1})\cdot (x_{2}\otimes y_{2}):=x_{1}x_{2}\otimes y_{1}y_{2}}$ ${\displaystyle E{\hat {\otimes }}_{\pi }F}$ ${\displaystyle E{\hat {\otimes }}_{\pi }F}$ ${\displaystyle \|\cdot \|_{\pi }}$

Negative statements

• A statement that is too analogous for spaces of continuous functions does not apply; for this one must use the injective tensor product . ${\displaystyle L^{1}(X,\Sigma ,\mu ){\hat {\otimes }}_{\pi }E\cong L^{1}(X,\Sigma ,\mu ,E)}$
• In general, the projective tensor product of reflexive spaces is not reflexive again. If the sequence space of the quadratically summable sequences with the unit vectors , then the closed subspace generated by the elements is from isometrically isomorphic to the sequence space of the absolutely summable sequences. Since the latter is not reflexive, it can not be reflexive either, although the Hilbert space is. ${\displaystyle \ell ^{2}}$ ${\displaystyle e_{n}}$ ${\displaystyle e_{n}\otimes e_{n}}$ ${\displaystyle \ell ^{2}{\hat {\otimes }}_{\pi }\ell ^{2}}$ ${\displaystyle \ell ^{1}}$ ${\displaystyle \ell ^{2}{\hat {\otimes }}_{\pi }\ell ^{2}}$ ${\displaystyle \ell ^{2}}$
• Disregarding trivial exceptions, projective tensor products of Hilbert spaces ( C ^* -algebras ) are not Hilbert spaces (C ^* -algebras), as is demonstrated by the example of the previous point. But there is a special Hilbert space tensor product that is also the starting point for tensor products of C * -algebras.

Local convex spaces

The construction of the projective tensor product can be generalized to the case of locally convex spaces.

definition

Let and be closed, absolutely convex zero neighborhoods in the locally convex vector spaces and . be the Minkowski functional of the absolutely convex hull of . The projective tensor or -Tensorprodukt is the Tensorproduktraum with the system of seminorms , wherein and through the closed, absolutely convex zero environments. ${\displaystyle U\subset E}$ ${\displaystyle V\subset F}$ ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle \pi _{U,V}}$ ${\displaystyle U\otimes V:=\{x\otimes y;x\in U,y\in V\}\subset E\otimes F}$ ${\displaystyle \pi }$ ${\displaystyle E\otimes _{\pi }F}$ ${\displaystyle \pi _{U,V}}$ ${\displaystyle U\subset E}$ ${\displaystyle V\subset F}$

If or denote the Minkowski functionals of or , the formula applies ${\displaystyle p_{U}}$ ${\displaystyle p_{V}}$ ${\displaystyle U}$ ${\displaystyle V}$

${\displaystyle \pi _{U,V}(z):=\inf\{\sum _{i=1}^{n}p_{U}(x_{i})\cdot p_{V}(y_{i});\,n\in {\mathbb {N} },x_{i}\in E,y_{i}\in F,z=\sum _{i=1}^{n}x_{i}\otimes y_{i}\}}$.

Hence this definition generalizes the projective tensor product of normalized spaces.

One can show that the topology explained in this way is the finest locally convex topology on the tensor product, which makes the natural bilinear mapping continuous. ${\displaystyle E\times F\rightarrow E\otimes F}$

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

Stability properties

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

• standardized spaces
• Metrizable locally convex spaces
• nuclear spaces
• Schwartz rooms
• quasi-normalizable spaces
• (DF) rooms
• gDF rooms ,

so also and belong to this class. ${\displaystyle E\otimes _{\pi }F}$ ${\displaystyle E{\hat {\otimes }}_{\pi }F}$

The projective tensor product of barreled spaces is generally not barreled again. But if and are metrizable and barreled, so is also metrizable and barreled. ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle E\otimes _{\pi }F}$

literature

• A. Grothendieck: Produits tensoriels topologiques et espaces nucléaires. Mem. Amer. Math. Soc., Volume 16, 1955.
• H. Jarchow: Locally Convex Spaces. Teubner, Stuttgart, 1981, ISBN 3-519-02224-9 .
• Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces , Springer-Verlag 2002, ISBN 1-85233-437-1 .
• R. Schatten: A theory of cross spaces. Annals of Mathematical Studies 26, Princeton, NJ 1950.

See also

• Vectorial measure : tensor products of spaces of measures

Individual evidence

1. ^ Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces , Springer-Verlag 2002, ISBN 1-85233-437-1 , Chapter 2.3: The Dual Space of ${\displaystyle X{\hat {\otimes }}_{\pi }Y}$
2. ↑ AY Helemskii: The homology of Banach and Topological Algebras. Kluwer Academic Publishers, 1989, ISBN 0-7923-0217-6 , Chapter II, Sentence 2.19
3. ^ Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces , Springer-Verlag 2002, ISBN 1-85233-437-1 , example 2.10