Let X,Y be Banach spaces. Then we have the following definition.
Definition 18.12.10We define for u ∈ X ⊗ Y
{ }
π(u) ≡ inf ∑ ∥xi∥∥yi∥ : u = ∑ xi ⊗ yi
i i
In this context, it is assumed that the elements of X ⊗ Y act on continuous bilinear forms. That is A isbilinear and
|A(x,y)| ≤ ∥A∥ ∥x∥∥y∥
When we write B
(X × Y ;V )
we mean the continuous ones. Here V is a normed vector space and theunderlying field for all is F.
Proposition 18.12.11π
(u )
is well defined and is a norm. Also, π
(x⊗ y)
=
∥x ∥
∥y∥
.
Proof:It is obviously well defined. Let λ ∈ F where F is the field of interest. Assume λ≠0 since
otherwise there is nothing to show. Then if u = ∑ixi⊗ yi.
∑ ∑
π(λu) ≤ ∥λxi∥∥yi∥ = |λ| ∥xi∥∥yi∥
i i
Then taking inf, it follows that
π(λu) ≤ |λ|π(u)
Then from what was just shown for arbitrary nonzero λ,
u(A) ≡ ∑ A(x ,y) ≤ ∑ ∥A∥∥x ∥∥y ∥ < (ε∕∥A ∥)∥A∥ = ε
i i i i i i
and since ε is arbitrary, this requires u
(A)
= 0. Since A is arbitrary, this requires u = 0.
Next is the very interesting equality that π
(x ⊗ y)
=
∥x∥
∥y∥
. It is obvious that
π(x ⊗y) ≤ ∥x∥∥y∥
because one way to write x ⊗ y is x ⊗ y. Let ϕ
(x)
=
∥x∥
,ψ
(y)
=
∥y∥
where
∥ϕ∥
,
∥ψ∥
= 1. Here
ϕ,ψ ∈ X′,Y′ respectively. You get them from the Hahn Banach theorem. Then consider the
continuous bilinear form A
(ˆx,ˆy)
≡ ϕ
(ˆx)
ψ
(ˆy)
. Say x ⊗ y = ∑ixi⊗ yi. There is a linear map
ψ ∈ℒ
(X ⊗ Y,V )
such that ψ
(ˆx ⊗ ˆy)
= A
(xˆ,yˆ)
. You just specify this on all things of the form
e ⊗ f where e ∈ E a Hamel basis for X and f ∈ F, a Hamel basis for Y . Then it must hold
for the linear span of these things which would yield the desired result. Hence, in particular,
| |
||∑ ||
∥x∥∥y∥ = ϕ (x) ψ(y) = |A(x,y)| = |ψ (x⊗ y)| = || ψ(xi ⊗ yi)||
| | | | i
||∑ || ||∑ || ∑
= || A (xi,yi)|| = || ϕ (xi)ψ (yi)|| ≤ ∥xi∥∥yi∥
i i i
It follows that on taking inf of both sides over all such representations of x ⊗ y that
∥x ∥∥y∥ ≤ π (x⊗ y).■
There is no difference if you replace X ⊗ Y with X1⊗ X2⊗
⋅⋅⋅
⊗ Xp. One modifies the definition as
follows.
Definition 18.12.12We define for u ∈ X1⊗ X2⊗
⋅⋅⋅
⊗ Xp
({ p ∥ ∥ )}
π(u) ≡ inf ∑ ∏ ∥∥xj∥∥ : u = ∑ x1 ⊗x2 ⊗ ⋅⋅⋅⊗ xp
( i j=1 i i i i i)
In this context, it is assumed that the elements of X1⊗ X2⊗
⋅⋅⋅
⊗ Xpact on continuous p linear forms.That is A is p linear and