Basic Concepts

After some remarks on bilinear mappings and the introduction of algebraic tensor products, the projective and injective tensor norms and some of their most important properties are studied in sections 3 – 6. Vector valued p–integrable functions, summability of sequences and averaging methods are treated in section 7 and section 8 – always from the perspective of tensor products. The remaining sections are devoted to the basics of the theory of operator ideals and integral and absolutely p-summing operators between Banach spaces, culminating in the so–called little Grothendieck theorem.

The reader should be familiar with some fundamental tools from Banach space theory, such as the Hahn–Banach theorem, the Mackey theorem / uniform boundedness principle, the weak– and weak–*–topologies, continuous linear operators and the open mapping and closed graph theorems. This clearly includes some simple knowledge about the classical Banach spaces C(K) and Lp, as well as the sequence spaces co and ℓp. In Appendix A some additional information about the structure theory of Banach spaces is collected.

1. Bilinear Mappings

This first section treats bilinear mappings and puts special emphasis on the fact that in many respects they behave differently than linear mappings, although they are intimately related with them.

1.1.

Let E, F, G be vector spaces over the same scalar field = or of the real or complex numbers. A mapping

: E × F → G

is called bilinear if the mappings

x Φ : F → G ? ? ? ? ? ? ? ? ? ? ? and ? Φ y : E → G ? ? ? ? ? ? y ⇝ Φ ( x , y ) ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? x ⇝ Φ ( x , y )

are linear for each x ∈ E and y ∈ F, in symbols : Φ ∈ Bil(E, F; G) if

x Φ ∈ L ( F , G ) ? and ? Φ y ∈ L ( E , G )

for each x ∈ E and y ∈ F. For simplicity: Bil(E, F) Bil(E, F; ). If E, F, G are normed spaces (or more generally: topological vector spaces), the set of continuous bilinear mappings E × F → G will be denoted by il(E, F; G) and il(E, F) if G = . From

( x , y ) − Φ ( x o , y o ) = Φ ( x − x o , y − y o ) + Φ ( x − x o , y o ) + Φ ( x o , y − y o )

the following result is easily deduced:

1.2.

A bilinear mapping Φ ∈ Bil(E, F; G) is separately continuous if all xΦ : F → G and Φy : E → G are continuous.

1.3.

Some examples of bilinear mappings:

(1)  For x′ ∈ E′ and y′ ∈ F′

x ′ ⊗ – y ′ ] ( x , y ) : = 〈 x ′ , x 〉 〈 y ′ , y 〉

defines a continuous bilinear form and x ′ ⊗ _ y ′ ‖ = ‖ x ′ ‖ ‖ y ′ ‖ . If xn′ and yn′ are in the unit balls and (λn) ∈ ℓ1, then

( x , y ) : = ∑ n = 1 ∞ λ n [ x ′ n ⊗ _ y ′ n ] ( x , y )

is well-defined and φ ‖ ≤ ∑ ∞ n = 1 | λ n | ; bilinear forms of this kind are called nuclear bilinear forms.

(2)  The evaluation map on the space (E, F) of continuous linear operators

( E , F ) × E → F ? ? ? ? ? ? ? ? ? ? ? ( T , x ) ⇝ T x

has norm 1 (if E and F are not trivial).

(3)  If E and F are finite dimensional, then all bilinear mappings E × F → G are continuous (use bases).

(4)  The convolution mapping

1 ( ℝ ) × L 1 ( ℝ ) → L 1 ( ℝ ) ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ( f , g ) ⇝ f * g

is bilinear.

(5)  Take the continuous functions on a compact space K and E a normed space. Then

( K ) × E → C ( K , E ) ? ? ? ? ? ? ? ? ? ( f , x ) ⇝ f ( ⋅ ) x

is bilinear.

1.4.

The mappings

i l ( E , F ) → L ( E , F * ) L ( E , F * ) → B i l ( E , F ) ? ? ? ? ? ? ? φ ⇕ ? ? ? ? ? ? ? L φ ? ? ? ? ? ? ? T ⇕ ? ? ? ? ? ? β T ? ? 〈 L φ x , y 〉 : = φ ( x , y ) β T ( x , y ) : = 〈 T x , y 〉

give isomorphisms of vector spaces and are inverse to each other. Since

φ ‖ = sup { | φ ( x , y ) | | x ∈ B E , y ∈ B F } sup { ‖ L φ x ‖ | x ∈ B E } = ‖ L φ ‖ ∈ [ 0 , ∞ ] ,

this isomorphism reduces for continuous bilinear forms to an isometry of normed spaces

il( E , F ) = 1 L ( E , F ′ ) ‖ L φ ‖ = ‖ φ ‖ ? and ? ‖ β T ‖ = ‖ T ‖.

This relationship is basic for the understanding of the ideas which will be presented in this book: the continuous bilinear forms on E × F are exactly the continuous operators E →F.′

1.5.

Since there is no Hahn-Banach theorem for operators, there is none for bilinear continuous forms in the following sense: Let G ⊂ E be a subspace and φ ∈ il(G, F); does there exist an extension ˜ ...