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Chapter 8: Problem 7

Let \(f, g\) be convex continuous functions on a Banach space \(X\) and assume that \(f\) is not Fr茅chet differentiable at \(x \in X .\) Show that \(f+g\) is not Fr茅chet differentiable at \(x\). Hint: By the assumption, \(f(x+h)+f(x-h)-2 f(x) \geq \varepsilon h\).

Short Answer

Expert verified
Since \( f(x+h) + f(x-h) - 2f(x) \geq \varepsilon \|h\| \), \( f+g \) is not Fr茅chet differentiable at \( x \).

Step by step solution

01

Define Fr茅chet differentiability

A function is Fr茅chet differentiable at a point if it can be approximated by a linear map in the neighborhood of that point. Formally, if there exists a bounded linear operator A such that \[ \lim_{\|h\| \to 0} \frac{\| f(x+h) - f(x) - A(h) \|}{\|h\|} = 0 \].
02

Consider the given condition for f

The condition given in the problem hints that \( f(x+h) + f(x-h) - 2f(x) \geq \varepsilon \|h\| \), for a certain \( \varepsilon > 0 \) and for all small \( h \). This implies that \( f(x+h) \) is not linearly approximable around \( x \).
03

Analyze the combined function f + g

We need to investigate the behavior of \(f+g\) around \(x\). Using the information from the previous steps, let's observe: \( (f+g)(x+h) + (f+g)(x-h) - 2(f+g)(x) \).
04

Express the combined function

We can rewrite the expression as: \( f(x+h) + g(x+h) + f(x-h) + g(x-h) - 2f(x) - 2g(x) \).
05

Apply the given condition

Since \( f(x+h) + f(x-h) - 2f(x) \geq \varepsilon \|h\| \), we can substitute this into our expression: \( \varepsilon \|h\| + g(x+h) + g(x-h) - 2g(x) \).
06

Recognize the behavior of g

Since \( g \) is convex and continuous, \( g(x+h) + g(x-h) - 2g(x) \geq 0 \).
07

Conclude non-differentiability

Combining both results, we get: \( (f+g)(x+h) + (f+g)(x-h) - 2(f+g)(x) \geq \varepsilon \|h\| \), which implies that \( f+g \) is also not linearly approximable around \( x\). Therefore, \( f+g \) is not Fr茅chet differentiable at \( x \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Banach Space
A Banach space is a complete normed vector space. This means it is a vector space equipped with a norm, and every Cauchy sequence in this space converges within the space.
In simpler terms, if you are working within a Banach space, you are assured that all the sequences that should converge, will indeed find their limit inside the space.
Here are some key points to remember about Banach spaces:
  • The space is linear, meaning it respects addition and scalar multiplication.
  • The norm assigns a length to each vector.
  • Completeness ensures that certain well-behaved sequences are bound to converge within the space.
Examples of Banach spaces include the spaces of continuous functions, sequences with bounded sums, and more.
Understanding Banach spaces is crucial in advanced mathematics, as they allow the generalization of many important concepts from finite-dimensional spaces to infinite-dimensional ones.
Convex Functions
Convex functions play a major role in optimization and analysis. A function is convex if the line segment between any two points on the graph of the function lies above or on the graph.
Mathematically, a function \(f\) is convex if for any points \(x, y\) in its domain and \(t \in [0, 1]\), the following holds:
\[ f(tx + (1-t)y) \leq t f(x) + (1-t) f(y) \]
This property is vital because it ensures there are no local minima other than the global minimum, making them easier to analyze and optimize.
Key properties of convex functions include:
  • If the second derivative of a function is non-negative everywhere, the function is convex.
  • Convex functions are closed under addition, meaning the sum of convex functions is also convex.
  • Convexity helps in analyzing the behavior of complex functions, especially when dealing with optimization problems.
Convex functions in Banach spaces retain many properties from finite-dimensional spaces, making them a powerful tool in mathematical analysis and applied mathematics.
Differentiation
Differentiation is the process of finding the derivative of a function. The derivative measures how a function changes as its input changes.
When talking about Fr茅chet differentiability, we're extending the idea of differentiation to functions between Banach spaces.
A function \(f: X \to Y\) between Banach spaces is Fr茅chet differentiable at a point \(x \in X\) if there exists a bounded linear operator \(A: X \to Y\) that closely approximates the change in \(f\).
Formally, \[ \lim_{\|h\| \to 0} \frac{\| f(x+h) - f(x) - A(h) \|}{\|h\|} = 0 \]
What this means is that near \(x\), the function \(f\) can be well-approximated by a linear map.
Important aspects of Fr茅chet differentiability include:
  • It's a stronger form of differentiability compared to G芒teaux differentiability, which only requires the directional derivative to exist.
  • It guarantees that the approximation by the linear operator is uniform in all directions.
  • In practical terms, it ensures smoother behaviors and helps in simplifying complex function analyses.
The concept is widely used in functional analysis and partial differential equations, where dealing with functions in infinite-dimensional spaces is common.

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Most popular questions from this chapter

Let \(C=\left\\{x=\left(x_{\gamma}\right) \in \ell_{2}(\Gamma) ; x_{\gamma} \geq 0\right.\) for all \(\left.\gamma \in \Gamma\right\\}\); that is, \(C\) is the positive cone in \(\ell_{2}(\Gamma)\). Let \(P\) assign to \(x \in \ell_{2}(\Gamma)\) its nearest point in \(C .\) Then \(P\) is a Lipschitz map (the previous exercise). Show that if \(\Gamma\) is uncountable, then \(P\) is nowhere G芒teaux differentiable. If \(\Gamma\) is infinite, then \(P\) is nowhere Fr茅chet differentiable. Note that Preiss proved that every real-valued Lipschitz function on \(\ell_{2}(\Gamma)\) is Fr茅chet differentiable on a dense set.

Let \(A\) be a subset of \(\mathbf{R}\) with \(\lambda(A)=0\) (Lebesgue measure). Follow the hint to show that there is a Lipschitz function \(f\) on \(\mathbf{R}\) not differentiable at points of \(A\) ([Zah]). Since there is a residual set of measure 0 in \(\mathbf{R}\), this shows that the set of differentiability points of a Lipschitz function need not be residual. Hint: \(([\mathrm{BeLi}])\) Take open sets \(G_{1} \supset G_{2} \supset \supset \ldots \supset A\) such that \(\lambda\left(G_{n}\right) \leq 2^{-n}\) and \(\lambda\left((a, b) \cap G_{n+1}\right) \leq \frac{b-a}{3}\) for every component \((a, b)\) of \(G_{n} .\) Put \(f_{n}(x)=\) \(\lambda\left((-\infty, x) \cap G_{n}\right)\) and \(f=\sum(-1)^{n+1} f .\) Then \(f\) is Lipschitz. If \(x \in \bigcap G_{n}\), then \(f\) is not differentiable at \(x .\) Indeed, let \(\left(\alpha_{n}, \beta_{n}\right)\) be components of \(G_{n}\) containing \(x\). Fix \(n\), define \(\gamma_{j}=\frac{f\left(\beta_{n}\right)-f\left(\alpha_{n}\right)}{\beta_{n}-\alpha_{n}}\), and observe that \(\left\\{\gamma_{j}\right\\}\) is decreasing, \(\gamma_{1}=\ldots=\gamma_{n}=1\), and \(\gamma_{n+1} \leq \frac{1}{3} .\) Thus if \(n\) is even, then \(\frac{f\left(\beta_{n}\right)-f\left(\alpha_{n}\right)}{\beta_{n}-\alpha_{n}}=1-1+\ldots-1+\gamma_{n+1}-\gamma_{n+2}+\ldots \leq \frac{1}{3}\), while if \(n\) is odd, \(\frac{f\left(\beta_{n}\right)-f\left(\alpha_{n}\right)}{\beta_{n}-\alpha_{n}}=1-1+\ldots+1-\gamma_{n+1}+\gamma_{n+2}-\ldots \geq \frac{2}{3}\)

Let a function \(f\) be Lipschitz on \(\mathbf{R}^{n}\) and G芒teaux differentiable at \(x\). Show that \(f\) is Fr茅chet differentiable at \(x .\) Is this also true for Lipschitz maps from \(\mathbf{R}^{n}\) to a Banach space \(Y ?\)

([Mon2]) Let \(X\) be a Banach space. The drop defined by \(x \in X \backslash B_{X}\) is the set \(D\left(x, B_{X}\right)=\operatorname{conv}\left(\\{x\\} \cup B_{X}\right) .\) The Banach space \(X\) is said to have the drop property if, given any closed set \(S \subset X\) such that \(S \cap B_{X}=\emptyset\), there exists \(s \in S\) such that \(D\left(s, B_{X}\right) \cap S=\\{x\\}\) (i) Let \(X\) be a Banach space. Show that, given a closed set \(A \subset X\) such that \(\operatorname{dist}\left(A, B_{X}\right)>0\), there exists \(a \in A\) such that \(D\left(a, B_{X}\right) \cap A=\\{a\\}\) ([Dan]). (ii) Prove that \(X\) has the drop property if and only if \(X\) is reflexive and has the Kadec property (that is, norm and weak convergent sequences in \(S_{X}\) are the same).

Let \(X\) be a separable reflexive Banach space. Show that extreme points of \(B_{X}\) are not all isolated in the norm topology.
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