Tensor product and infinite dimensional vector space

In summary, the statement holds true that every linear map between two vector spaces can be expanded using the tensor product, including in the case of infinite dimensional vector spaces. This can be observed in the expansion of the metric in General Relativity. An example of this can be found in the outer product definition in Wikipedia.
  • #1
ivl
27
0
Hi all,

it is of course true that every linear map between two vector spaces can be expanded by means of the tensor product.

For instance, the metric in General Relativity (mapping covectors to vectors) can be expanded as

[itex]g=\sum_{i,j}g^{ij}e_{i}\otimes e_{j}[/itex].

However, does this statement hold true when the linear operator maps between infinite dimensional vector spaces?

Any help is very much appreciated!
 
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  • #2
ivl said:
Hi all,

it is of course true that every linear map between two vector spaces can be expanded by means of the tensor product.

For instance, the metric in General Relativity (mapping covectors to vectors) can be expanded as

[itex]g=\sum_{i,j}g^{ij}e_{i}\otimes e_{j}[/itex].

However, does this statement hold true when the linear operator maps between infinite dimensional vector spaces?

Any help is very much appreciated!

I found an answer in wikipedia, see:

http://en.wikipedia.org/wiki/Outer_product

towards the end of the section "Definition (abstract)".
 

FAQ: Tensor product and infinite dimensional vector space

What is a tensor product?

The tensor product is a mathematical operation that combines two vector spaces to create a new vector space. It is denoted by the symbol ⊗ and is used to represent the relationship between two vector spaces. The result of a tensor product is a vector space that contains all possible combinations of the basis vectors from the original two spaces.

How is the tensor product different from the cross product?

The tensor product and cross product are two different operations in mathematics. The tensor product is used to combine vector spaces, while the cross product is used to find a vector that is perpendicular to two given vectors. Additionally, the tensor product can be applied to any vector spaces, while the cross product is only defined for three-dimensional vectors.

What is an infinite dimensional vector space?

An infinite dimensional vector space is a vector space that contains an infinite number of basis vectors. This means that the vectors in the space are not finite and cannot be represented by a finite set of coordinates. Infinite dimensional vector spaces are commonly used in functional analysis and other areas of mathematics.

How is the tensor product used in quantum mechanics?

In quantum mechanics, the tensor product is used to represent the relationship between two quantum states. This allows for the combination of different states to create a new state that represents a composite system. The tensor product is also used in the mathematical formulation of entanglement and other quantum phenomena.

What are some applications of the tensor product in engineering?

The tensor product has many applications in engineering, particularly in the fields of control systems, signal processing, and image processing. It is used to represent multidimensional signals and systems, as well as to model the interactions between different components in a system. The tensor product is also used in machine learning and artificial intelligence algorithms for data representation and manipulation.

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