Quantum Mechanics for Mathematicians

• 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}.