- There are natural isomorphisms
V⊗W'W⊗V
and
U⊗(V⊗W)'(U⊗V)⊗W
for vector spacesU,V,W
- Given a linear operatorAonVand another linear operatorBonW, we
can define a linear operatorA⊗BonV⊗Wby
(A⊗B)(v⊗w) =Av⊗Bw
forv∈V,w∈W.
With respect to the basesej,fkofV andW,Awill be a (dimV) by
(dimV) matrix,Bwill be a (dimW) by (dimW) matrix andA⊗Bwill
be a (dimV)(dimW) by (dimV)(dimW) matrix (which can be thought
of as a (dimV) by (dimV) matrix of blocks of size (dimW)).
- One often wants to consider tensor products of vector spaces and dual
vector spaces. An important fact is that there is an isomorphism between
the tensor productV∗⊗W and linear maps fromVtoW. This is given
by identifyingl⊗w(l∈V∗,w∈W) with the linear map
v∈V→l(v)w∈W
- Given the motivation in terms of functions on a product of sets, for func-
tion spaces in general we should have an identification of the tensor prod-
uct of function spaces with functions on the product set. For instance, for
square-integrable functions onRwe expect
L^2 (R)⊗L^2 (R)⊗···⊗L^2 (R)
︸ ︷︷ ︸
n times
=L^2 (Rn) (9.1)
ForV a real vector space, its complexificationVC(see section 5.5) can be
identified with the tensor product
VC=V⊗RC
Here the notation⊗Rindicates a tensor product of two real vector spaces:V
of dimension dimVwith basis{e 1 ,e 2 ,...,edimV}andC=R^2 of dimension 2
with basis{ 1 ,i}.