Differentiable vector-valued functions from Euclidean space: Difference between revisions

Revision as of 01:36, 15 February 2021

Revision as of 01:36, 15 February 2021

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=== Identification as a tensor product ===

Suppose henceforth that <math>Y</math> is a Hausdorff [[topological vector space]].

Given a function <math>f \in C^k(\Omega)</math> and a vector <math>y \in Y,</math> let <math>f \otimes y</math> denote the map <math>f \otimes y : \Omega \to Y</math> defined by <math>(f \otimes y)(p) = f(p) y.</math>

This defines a bilinear map <math>\otimes : C^k(\Omega) \times Y \to C^k(\Omega;Y)</math> into the space of functions whose image is contained in a finite-dimensional vector subspace of <math>Y</math>;