# Sobolev inequalities for Banach-valued functions

To what extend do the various different Sobolev inequalites hold if I replace the usual target space $\mathbb R$ by an arbitrary Banach space, the notion of derivative by Frechét derivative and the usual real-valued integral by the Bochner integral (correspondingly adapting the definition of weak derivative) ?

• Typically, Hahn-Banach helps for such generalizations. – Jochen Wengenroth Mar 22 '18 at 15:05
• Could you give an example on how one would make use of Hahn-Banach in this situation ? I am frankly quite surprised that one can apply this Theorem here. – Berni Waterman Mar 22 '18 at 15:10