# G\^Ateaux-Differentiability of Convex Functions in Infinite Dimension

## TL;DR

This paper extends the classical equivalence between Gâteaux differentiability and the existence of partial derivatives for convex functions from finite-dimensional spaces to certain infinite-dimensional Banach spaces with a Schauder basis.

## Contribution

It generalizes the finite-dimensional result to infinite-dimensional Banach spaces, establishing conditions under which Gâteaux differentiability implies the existence of partial derivatives.

## Key findings

- Gâteaux differentiability is equivalent to partial derivatives in Banach spaces with a Schauder basis.
- The result extends classical finite-dimensional convex analysis to infinite-dimensional settings.
- Provides a foundation for analyzing convex functions in infinite-dimensional analysis.

## Abstract

It is well known that in $R^n$ , G{\^a}teaux (hence Fr{\'e}chet) differ-entiability of a convex continuous function at some point is equivalent to the existence of the partial derivatives at this point. We prove that this result extends naturally to certain infinite dimensional vector spaces, in particular to Banach spaces having a Schauder basis.