Banach space
Tensor product

Let X {\displaystyle X} and Y {\displaystyle Y} be two K {\displaystyle \mathbb {K} } -vector spaces. The tensor product X ⊗ Y {\displaystyle X\otimes Y} of X {\displaystyle X} and Y {\displaystyle Y} is a K {\displaystyle \mathbb {K} } -vector space Z {\displaystyle Z} with a bilinear mapping T: X × Y → Z {\displaystyle T:X\times Y\to Z} which has the following universal property:

The image under T {\displaystyle T} of a couple (x, y) {\displaystyle (x,y)} in X × Y {\displaystyle X\times Y} is denoted by x ⊗ y, {\displaystyle x\otimes y,} and called a simple tensor. Every element z {\displaystyle z} in X ⊗ Y {\displaystyle X\otimes Y} is a finite sum of such simple tensors.

There are various norms that can be placed on the tensor product of the underlying vector spaces, amongst others the projective cross norm and injective cross norm introduced by A. Grothendieck in 1955.

In general, the tensor product of complete spaces is not complete again. When working with Banach spaces, it is customary to say that the projective tensor product of two Banach spaces X {\displaystyle X} and Y {\displaystyle Y} is the completion X ⊗ ^ π Y {\displaystyle X{\widehat {\otimes }}_{\pi }Y} of the algebraic tensor product X ⊗ Y {\displaystyle X\otimes Y} equipped with the projective tensor norm, and similarly for the injective tensor product X ⊗ ^ ε Y. {\displaystyle X{\widehat {\otimes }}_{\varepsilon }Y.} Grothendieck proved in particular that

C (K) ⊗ ^ ε Y ≃ C (K, Y), L 1 () ⊗ ^ π Y ≃ L 1 (, Y), {\displaystyle {\begin{aligned}C(K){\widehat {\otimes }}_{\varepsilon }Y&\simeq C(K,Y),\\L^{1}(){\widehat {\otimes }}_{\pi }Y&\simeq L^{1}(,Y),\end{aligned}}} where K {\displaystyle K} is a compact Hausdorff space, C (K, Y) {\displaystyle C(K,Y)} the Banach space of continuous functions from K {\displaystyle K} to Y {\displaystyle Y} and L 1 (, Y) {\displaystyle L^{1}(,Y)} the space of Bochner-measurable and integrable functions from {\displaystyle } to Y, {\displaystyle Y,} and where the isomorphisms are isometric. The two isomorphisms above are the respective extensions of the map sending the tensor f ⊗ y {\displaystyle f\otimes y} to the vector-valued function s ∈ K → f (s) y ∈ Y. {\displaystyle s\in K\to f(s)y\in Y.}
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