Question

Let \(x_{i}\) be vectors in a Banach space \(X\) such that \(\overline{\operatorname{span}}\left\\{x_{i}\right\\}=X\). Assume that \(f\) is a continuous convex function on \(X\) such that, at all points of \(X\), all directional derivatives in the directions of \(\left\\{x_{i}\right\\}\) exist. Is \(f\) Gâteaux differentiable on \(X ?\)

Short answer

Yes, \( f \) is Gâteaux differentiable on \( X \).

Step-by-step solution

Understand the Given Information

The exercise provides a Banach space \( X \) and vectors \( x_i \) in \( X \) such that their span is dense in \( X \) (i.e., \( \overline{\operatorname{span}}\{x_i\} = X \)). A continuous convex function \( f \) defined on \( X \) with existing directional derivatives in the directions of \( \{x_i\} \) at every point in \( X \) is given. The goal is to determine if \( f \) is Gâteaux differentiable on \( X \).

Review Definitions

Recall that a function \( f \) is Gâteaux differentiable at a point \( x \in X \) if the limit \[ \lim_{t \to 0} \frac{f(x + th) - f(x)}{t} \] exists for all directions \( h \in X \). We need to check if this condition is satisfied for \( f \) given the provided assumptions.

Consider Convexity and Continuity

Convexity ensures that the directional derivatives of \( f \) in any direction are monotonically increasing or decreasing. Since \( f \) is also continuous, we know it maps bounded sets to bounded sets in \( X \).

Use Directional Derivatives

Given that directional derivatives \( \frac{d}{dt} f(x + t x_i) \) exist for every \( x \in X \) and \( x_i \), and span \( \{x_i\} \) is dense in \( X \), it implies that we can approximate any direction \( h \) by a sequence of directions in \( \operatorname{span}\{x_i\} \).

Evaluate Gâteaux Differentiability

Since \( f \) is continuous and convex, and has existing directional derivatives in a dense set of directions, \( f \) is Gâteaux differentiable at all points in \( X \). The density of \( \{x_i\} \) ensures that the limit definition for Gâteaux differentiability holds in all directions.

Key concepts

Convex Function

A function is defined as convex if, for any two points within its domain, the line segment connecting these points lies above or on the graph of the function. Mathematically, a function \(f: X \to \text{R}\) is convex if for any \(x, y \in X\) and \(\lambda \in [0,1]\), we have:

\[ f(\lambda x + (1 - \lambda) y) \leq \lambda f(x) + (1 - \lambda) f(y) \]

Convex functions have some very nice properties:

• Their local minima are also global minima.
• Their epigraphs (the set of points above the graph) are convex sets.

Convexity plays a significant role in optimization problems and analysis. Continuous convex functions maintain boundedness over bounded sets, making them easier to handle analytically.

Gâteaux Differentiability

A function \(f\) is said to be Gâteaux differentiable at a point \(x \in X\) if it has a linear approximation along any direction \(h \in X\). Precisely, \(f\) is Gâteaux differentiable at \(x\) if the following limit exists for all directions \(h\):

\[ \lim_{t \to 0} \frac{f(x + th) - f(x)}{t} \]

This definition is a generalization of the usual derivative to infinite-dimensional spaces. Unlike the standard derivative, Gâteaux derivative only guarantees existence in specific directions, not all directions simultaneously. However, in the case of convex and continuous functions, Gâteaux differentiability is rather typical due to the smoothness and predictability imposed by convexity.

Directional Derivatives

Directional derivatives extend the concept of ordinary derivatives to the case where the direction of differentiation varies. For a function \(f\), the directional derivative at a point \(x\) in the direction \(v\) is defined as:

\[ \frac{d}{dt} f(x + tv) \bigg|_{t=0} \]

This derivative quantifies how the function \(f\) changes as one moves infinitesimally in the direction of \(v\). They provide critical insights into the local behavior of \(f\):

• If \(f\) is differentiable, the directional derivative in any direction exists and is given by the dot product of the gradient with the direction vector.
• Existence in a dense set of directions often leads to broader differentiability properties, as in our specific exercise scenario.

Directional derivatives are instrumental in optimization and variational analysis problems.

Dense Span

In mathematical analysis, a set \(\{x_i\}\) in a Banach space \(X\) has a dense span if the closure of its linear span equals the entire space, denoted as \(\overline{\operatorname{span}}\{x_i\}=X\). This means that any element of \(X\) can be approximated arbitrarily closely by finite linear combinations of \(x_i\):

• Dense sets provide a sufficient framework to build other elements of the space.
• In practical terms, dense spans allow us to generalize results from a finite, manageable set of vectors to the whole infinite-dimensional space.

This concept is crucial in our exercise because it allows us to use the available directional derivatives in a dense set of directions to establish the Gâteaux differentiability of \(f\) in the entire Banach space.