Linear algebra MOC

-tensor product of vector spaces

Let be vector spaces over . The tensor product is a vector space which allows one to treat -bilinear maps from as -linear maps from , as ensured by the Universal property. See Tensor product of modules over a commutative ring for a direct generalization.

Universal property

The tensor product is uniquely characterised by the following universal property:

If are vector spaces over and is a bilinear map there exists a unique linear map such that .

Proof

proof Let be a bilinear map.

Finite dimensional characterization

The tensor product of vector spaces over a field is the vector space of bilinear forms , equipped with a bilinear¹ map such that linalg

for , , , .²³ It follows that if and are bases of and respectively, then defines a basis for the tensor product space . We call

a type Tensor⁴

Warning

This characterization probably requires and hence finite dimensions.

Hilbert spaces

If are finite-dimensional Hilbert spaces. then the tensor product is a Hilbert space carrying the unique inner product given by

Then if and are orthonormal bases of respectively, is an orthonormal basis of .

Further characterizations

As quotient

One may construct the tensor product as a quotient space of the free module generated by formal products of vectors in and .

Properties

See also

• Tensor product of linear maps (functor)
• Tensor product of group representations
• Tensor algebra
• Tensor

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Footnotes

1. Simon defines these as “biantilinear” maps , which is of course completely equivalent. ↩

2. 1996, Representations of finite and compact groups, §II.5, p. 29 ↩

3. 2015. An Introduction to Tensors and Group Theory for Physicists, §3.4, p.70 ↩

4. Authors vary on the order of the tensor type, cf. @jeevanjeeIntroductionTensorsGroup2015 with @emamCovariantPhysicsClassical2021 (I use the convention of the latter, also aligns with Wikipedia) ↩