Smooth extensions of functions on separable Banach spaces

TL;DR

This paper proves that any $C^{1}$-smooth function defined on a closed subspace of a Banach space with a separable dual can be extended to a $C^{1}$-smooth function on the entire space.

Contribution

It establishes the existence of $C^{1}$ smooth extensions for functions on subspaces of Banach spaces with separable duals, generalizing previous extension results.

Findings

Existence of $C^{1}$ extensions for functions on subspaces

Extension applies to Banach spaces with separable duals

Provides a constructive method for smooth extensions

Abstract

Let $X$ be a Banach space with a separable dual $X^{*}$. Let $Y\subset X$ be a closed subspace, and $f:Y\to\mathbb{R}$ a $C^{1}$-smooth function. Then we show there is a $C^{1}$ extension of $f$ to $X$.