Linear Algebra and Analysis

18.12.1 The Norm In Tensor Product Space

Let X,Y be Banach spaces. Then we have the following definition.

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 = ∑ _ix_i ⊗ y_i.

∑ ∑ 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,

π (u) = π(λ−1λu) ≤ -1-π(λu) |λ|

and so

π ≤ π. Hence π = π.

Let u = ∑ _ix_i ⊗ y_i,v = ∑ _j

_j ⊗ŷ_j such that

∑ ∑ π (u)+ ε > ∥xi∥∥yi∥ , π(v)+ ε > ∥xˆj∥ ∥ˆyj∥ . i j

Then

∑ ∑ π(u +v) ≤ ∥xi∥ ∥yi∥ + ∥ˆxj∥ ∥ˆyj∥ ≤ π(u)+ π (v) + 2ε i j

Since ε is arbitrary, π

≤ π + π.

Now suppose π

= 0. Does it follow that u = 0 for all A ∈ B? Let A ∈ B and let u = ∑ _ix_i ⊗ y_i such that

∑ ε∕ ∥A∥ = π (u)+ ε∕∥A∥ > ∥xi∥∥yi∥ i

Then

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

= 0. Since A is arbitrary, this requires u = 0.

Next is the very interesting equality that π

= . It is obvious that

π(x ⊗y) ≤ ∥x∥∥y∥

because one way to write x ⊗ y is x ⊗ y. Let ϕ

= ,ψ = where , = 1. Here ϕ,ψ ∈ X^′,Y ^′ respectively. You get them from the Hahn Banach theorem. Then consider the continuous bilinear form A ≡ ϕψ. Say x ⊗ y = ∑ _ix_i ⊗ y_i. There is a linear map ψ ∈ℒ such that ψ = A. 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,

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 ⊗

⊗ X_p. One modifies the definition as follows.

Recall Corollary 18.12.9

X1 × ⋅⋅⋅× Xp p ψ ⊗ ↓ ↘↺ ˆψ X1 ⊗X2 ⊗ ⋅⋅⋅⊗ Xp −−→ V

Is it the case that

is continuous? Letting a ∈ X₁ ⊗ X₂ ⊗⊗ X_p, does it follow that _V ≤X₁⊗X₂⊗

⋅⋅⋅

⊗X_p?

Let a = ∑ _ix_i¹ ⊗ x_i² ⊗

⊗ x_i^p. Then

Then taking the inf over all such representations of a, it follows that ≤X₁⊗X₂⊗

⋅⋅⋅

⊗X_p. This proves the following theorem.