Differentiable vector-valued functions from Euclidean space: Difference between revisions
Content deleted Content added
Fix |
m Fix |
||
Line 105:
Given <math>\phi \in C^k(\Omega \times \Delta),</math> for any <math>y \in \Delta</math> let <math>\phi_y : \Omega \to \mathbb{F}</math> be defined by <math>\phi_y(x) = \phi(x, y)</math> and let <math>I_k(\phi) : \Delta \to C^k(\Omega)</math> be defined by <math>I_k(\phi)(y) := \phi_y.</math>
Then
:<math>I_\infty : C^\infty(\Omega \times \Delta) \to C^\infty(\Delta; C^\infty(\Omega))</math>
is a surjective isomorphism of TVSs. Furthermore, its restriction
:<math>I_{\infty}\big\vert_{C_c^{\infty}\left(\Omega \times \Delta\right)} : C_c^\infty(\Omega \times \Delta) \to C_c^\infty\left(\Delta; C_c^\infty(\Omega)\right)</math>
| |||