# Topological tensor product

Last updated

In mathematics, there are usually many different ways to construct a topological tensor product of two topological vector spaces. For Hilbert spaces or nuclear spaces there is a simple well-behaved theory of tensor products (see Tensor product of Hilbert spaces), but for general Banach spaces or locally convex topological vector spaces the theory is notoriously subtle.

## Motivation

One of the original motivations for topological tensor products ${\displaystyle {\hat {\otimes }}}$ is the fact that tensor products of the spaces of smooth functions on ${\displaystyle \mathbb {R} ^{n}}$ do not behave as expected. There is an injection

${\displaystyle C^{\infty }(\mathbb {R} ^{n})\otimes C^{\infty }(\mathbb {R} ^{m})\hookrightarrow C^{\infty }(\mathbb {R} ^{n+m})}$

but this is not an isomorphism. For example, the function ${\displaystyle f(x,y)=e^{xy}}$ cannot be expressed as a finite linear combination of smooth functions in ${\displaystyle C^{\infty }(\mathbb {R} _{x})\otimes C^{\infty }(\mathbb {R} _{y}).}$ [1] We only get an isomorphism after constructing the topological tensor product; i.e.,

${\displaystyle C^{\infty }(\mathbb {R} ^{n})\mathop {\hat {\otimes }} C^{\infty }(\mathbb {R} ^{m})\cong C^{\infty }(\mathbb {R} ^{n+m})}$

This article first details the construction in the Banach space case. ${\displaystyle C^{\infty }(\mathbb {R} ^{n})}$ is not a Banach space and further cases are discussed at the end.

## Tensor products of Hilbert spaces

The algebraic tensor product of two Hilbert spaces A and B has a natural positive definite sesquilinear form (scalar product) induced by the sesquilinear forms of A and B. So in particular it has a natural positive definite quadratic form, and the corresponding completion is a Hilbert space AB, called the (Hilbert space) tensor product of A and B.

If the vectors ai and bj run through orthonormal bases of A and B, then the vectors aibj form an orthonormal basis of AB.

## Cross norms and tensor products of Banach spaces

We shall use the notation from ( Ryan 2002 ) in this section. The obvious way to define the tensor product of two Banach spaces A and B is to copy the method for Hilbert spaces: define a norm on the algebraic tensor product, then take the completion in this norm. The problem is that there is more than one natural way to define a norm on the tensor product.

If A and B are Banach spaces the algebraic tensor product of A and B means the tensor product of A and B as vector spaces and is denoted by ${\displaystyle A\otimes B}$. The algebraic tensor product ${\displaystyle A\otimes B}$ consists of all finite sums

${\displaystyle x=\sum _{i=1}^{n}a_{i}\otimes b_{i}}$

where ${\displaystyle n}$ is a natural number depending on ${\displaystyle x}$ and ${\displaystyle a_{i}\in A}$ and ${\displaystyle b_{i}\in B}$ for ${\displaystyle i=1,\ldots ,n}$.

When A and B are Banach spaces, a cross normp on the algebraic tensor product ${\displaystyle A\otimes B}$ is a norm satisfying the conditions

${\displaystyle p(a\otimes b)=\|a\|\|b\|,}$
${\displaystyle p'(a'\otimes b')=\|a'\|\|b'\|.}$

Here a′ and b′ are in the topological dual spaces of A and B, respectively, and p′ is the dual norm of p. The term reasonable crossnorm is also used for the definition above.

There is a cross norm ${\displaystyle \pi }$ called the projective cross norm, given by

${\displaystyle \pi (x)=\inf \left\{\sum _{i=1}^{n}\|a_{i}\|\|b_{i}\|:x=\sum _{i}a_{i}\otimes b_{i}\right\}}$

where ${\displaystyle x\in A\otimes B}$.

It turns out that the projective cross norm agrees with the largest cross norm (( Ryan 2002 ), proposition 2.1).

There is a cross norm ${\displaystyle \varepsilon }$ called the injective cross norm, given by

${\displaystyle \varepsilon (x)=\sup\{|(a'\otimes b')(x)|:a'\in A',b'\in B',\|a'\|=\|b'\|=1\}}$

where ${\displaystyle x\in A\otimes B}$. Here A′ and B′ mean the topological duals of A and B, respectively.

Note hereby that the injective cross norm is only in some reasonable sense the "smallest".

The completions of the algebraic tensor product in these two norms are called the projective and injective tensor products, and are denoted by ${\displaystyle A\operatorname {\hat {\otimes }} _{\pi }B}$ and ${\displaystyle A\operatorname {\hat {\otimes }} _{\varepsilon }B.}$

When A and B are Hilbert spaces, the norm used for their Hilbert space tensor product is not equal to either of these norms in general. Some authors denote it by σ, so the Hilbert space tensor product in the section above would be ${\displaystyle A\operatorname {\hat {\otimes }} _{\sigma }B.}$

A uniform crossnorm α is an assignment to each pair ${\displaystyle (X,Y)}$ of Banach spaces of a reasonable crossnorm on ${\displaystyle X\otimes Y}$ so that if ${\displaystyle X,W,Y,Z}$ are arbitrary Banach spaces then for all (continuous linear) operators ${\displaystyle S:X\to W}$ and ${\displaystyle T:Y\to Z}$ the operator ${\displaystyle S\otimes T:X\otimes _{\alpha }Y\to W\otimes _{\alpha }Z}$ is continuous and ${\displaystyle \|S\otimes T\|\leq \|S\|\|T\|.}$ If A and B are two Banach spaces and α is a uniform cross norm then α defines a reasonable cross norm on the algebraic tensor product ${\displaystyle A\otimes B.}$ The normed linear space obtained by equipping ${\displaystyle A\otimes B}$ with that norm is denoted by ${\displaystyle A\otimes _{\alpha }B.}$ The completion of ${\displaystyle A\otimes _{\alpha }B,}$ which is a Banach space, is denoted by ${\displaystyle A\operatorname {\hat {\otimes }} _{\alpha }B.}$ The value of the norm given by α on ${\displaystyle A\otimes B}$ and on the completed tensor product ${\displaystyle A\operatorname {\hat {\otimes }} _{\alpha }B}$ for an element x in ${\displaystyle A\operatorname {\hat {\otimes }} _{\alpha }B}$ (or ${\displaystyle A\otimes _{\alpha }B}$) is denoted by ${\displaystyle \alpha _{A,B}(x)}$ or ${\displaystyle \alpha (x).}$

A uniform crossnorm ${\displaystyle \alpha }$ is said to be finitely generated if, for every pair ${\displaystyle (X,Y)}$ of Banach spaces and every ${\displaystyle u\in X\otimes Y}$,

${\displaystyle \alpha (u;X\otimes Y)=\inf\{\alpha (u;M\otimes N):\dim M,\dim N<\infty \}.}$

A uniform crossnorm ${\displaystyle \alpha }$ is cofinitely generated if, for every pair ${\displaystyle (X,Y)}$ of Banach spaces and every ${\displaystyle u\in X\otimes Y}$,

${\displaystyle \alpha (u)=\sup\{\alpha ((Q_{E}\otimes Q_{F})u;(X/E)\otimes (Y/F)):\dim X/E,\dim Y/F<\infty \}.}$

A tensor norm is defined to be a finitely generated uniform crossnorm. The projective cross norm ${\displaystyle \pi }$ and the injective cross norm ${\displaystyle \varepsilon }$ defined above are tensor norms and they are called the projective tensor norm and the injective tensor norm, respectively.

If A and B are arbitrary Banach spaces and α is an arbitrary uniform cross norm then

${\displaystyle \varepsilon _{A,B}(x)\leq \alpha _{A,B}(x)\leq \pi _{A,B}(x).}$

## Tensor products of locally convex topological vector spaces

The topologies of locally convex topological vector spaces ${\displaystyle A}$ and ${\displaystyle B}$ are given by families of seminorms. For each choice of seminorm on ${\displaystyle A}$ and on ${\displaystyle B}$ we can define the corresponding family of cross norms on the algebraic tensor product ${\displaystyle A\otimes B,}$ and by choosing one cross norm from each family we get some cross norms on ${\displaystyle A\otimes B,}$ defining a topology. There are in general an enormous number of ways to do this. The two most important ways are to take all the projective cross norms, or all the injective cross norms. The completions of the resulting topologies on ${\displaystyle A\otimes B}$ are called the projective and injective tensor products, and denoted by ${\displaystyle A\otimes _{\gamma }B}$ and ${\displaystyle A\otimes _{\lambda }B.}$ There is a natural map from ${\displaystyle A\otimes _{\gamma }B}$ to ${\displaystyle A\otimes _{\lambda }B.}$

If ${\displaystyle A}$ or ${\displaystyle B}$ is a nuclear space then the natural map from ${\displaystyle A\otimes _{\gamma }B}$ to ${\displaystyle A\otimes _{\lambda }B}$ is an isomorphism. Roughly speaking, this means that if ${\displaystyle A}$ or ${\displaystyle B}$ is nuclear, then there is only one sensible tensor product of ${\displaystyle A}$ and ${\displaystyle B}$. This property characterizes nuclear spaces.

## Related Research Articles

In mathematics, more specifically in functional analysis, a Banach space is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space.

In mathematics, any vector space V has a corresponding dual vector space consisting of all linear forms on V, together with the vector space structure of pointwise addition and scalar multiplication by constants.

In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" in the real world. A norm is a real-valued function defined on the vector space that is commonly denoted and has the following properties:

1. It is nonnegative, that is for every vector x, one has
2. It is positive on nonzero vectors, that is,
3. For every vector x, and every scalar one has
4. The triangle inequality holds; that is, for every vectors x and y, one has

In mathematics, a product is the result of multiplication, or an expression that identifies factors to be multiplied. For example, 30 is the product of 6 and 5, and is the product of and .

In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.

In mathematics, the tensor productVW of two vector spaces V and W is a vector space, endowed with a bilinear map from the Cartesian product V × W to VW. This bilinear map is universal in the sense that, for every vector space X, the bilinear maps from V × W to X are in one to one correspondence with the linear maps from VW to X.

Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions than classical solutions, or appropriate classical solutions may not exist. Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are distributions, such as the Dirac delta function.

In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, making it an example of a coproduct. Contrast with the direct product, which is the dual notion.

In functional analysis, an F-space is a vector space V over the real or complex numbers together with a metric d : V × V → ℝ so that

1. Scalar multiplication in V is continuous with respect to d and the standard metric on ℝ or ℂ.
2. Addition in V is continuous with respect to d.
3. The metric is translation-invariant; i.e., d(x + a, y + a) = d(x, y) for all x, y and a in V
4. The metric space (V, d) is complete.

In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space (TVS) such that the canonical evaluation map from X into its bidual is an isomorphism of TVSs. Since a normable TVS is reflexive if and only if it is semi-reflexive, every normed space X is reflexive if and only if the canonical evaluation map from X into its bidual is surjective; in this case the normed space is necessarily also a Banach space. Note that in 1951, R. C. James discovered a non-reflexive Banach space that is isometrically isomorphic to its bidual.

In mathematics, the complexification of a vector space V over the field of real numbers yields a vector space V ℂ over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include their scaling ("multiplication") by complex numbers. Any basis for V may also serve as a basis for V ℂ over the complex numbers.

In mathematics, a nuclear space is a topological vector space with many of the good properties of finite-dimensional vector spaces. The topology on them can be defined by a family of seminorms whose unit balls decrease rapidly in size. Vector spaces whose elements are "smooth" in some sense tend to be nuclear spaces; a typical example of a nuclear space is the set of smooth functions on a compact manifold.

In mathematics, and in particular functional analysis, the tensor product of Hilbert spaces is a way to extend the tensor product construction so that the result of taking a tensor product of two Hilbert spaces is another Hilbert space. Roughly speaking, the tensor product is the metric space completion of the ordinary tensor product. This is an example of a topological tensor product. The tensor product allows Hilbert spaces to be collected into a symmetric monoidal category.

In mathematics, a Schauder basis or countable basis is similar to the usual (Hamel) basis of a vector space; the difference is that Hamel bases use linear combinations that are finite sums, while for Schauder bases they may be infinite sums. This makes Schauder bases more suitable for the analysis of infinite-dimensional topological vector spaces including Banach spaces.

Given a Hilbert space with a tensor product structure a product numerical range is defined as a numerical range with respect to the subset of product vectors. In some situations, especially in the context of quantum mechanics product numerical range is known as local numerical range

The strongest locally convex topological vector space (TVS) topology on , the tensor product of two locally convex TVSs, making the canonical map continuous is called the projective topology or the π-topology. When X ⊗ Y is endowed with this topology then it is denoted by and called the projective tensor product of X and Y.

In mathematics, the injective tensor product of two topological vector spaces (TVSs) was introduced by Alexander Grothendieck and was used by him to define nuclear spaces. An injective tensor product is in general not necessarily complete, so its completion is called the completed injective tensor products. Injective tensor products have applications outside of nuclear spaces. In particular, as described below, up to TVS-isomorphism, many TVSs that are defined for real or complex valued functions, for instance, the Schwartz space or the space of continuously differentiable functions, can be immediately extended to functions valued in a Hausdorff locally convex TVS Y without any need to extend definitions from real/complex-valued functions to Y-valued functions.

In mathematics, nuclear operators are an important class of linear operators introduced by Alexander Grothendieck in his doctoral dissertation. Nuclear operators are intimately tied to the projective tensor product of two topological vector spaces (TVSs).

In the mathematical discipline of functional analysis, it is possible to generalize the notion of derivative to infinite dimensional topological vector spaces (TVSs) in multiple ways. But when the domain of TVS-value functions is a subset of finite-dimensional Euclidean space then the number of generalizations of the derivative is much more limited and derivatives are more well behaved. This article presents the theory of k-times continuously differentiable functions on an open subset of Euclidean space , which is an important special case of differentiation between arbitrary TVSs. All vector spaces will be assumed to be over the field where is either the real numbers or the complex numbers

This is a glossary for the terminology in a mathematical field of functional analysis.

## References

1. "What is an example of a smooth function in C∞(R2) which is not contained in C∞(R)⊗C∞(R)".
• Ryan, R.A. (2002), Introduction to Tensor Products of Banach Spaces, New York: Springer.
• Grothendieck, A. (1955), "Produits tensoriels topologiques et espaces nucléaires", Memoirs of the American Mathematical Society, 16.