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Chapter 8: Problem 41

Show that the nonlinear operator \(\varphi\) from \(L_{2}[0,1]\) into \(L_{2}[0,1]\) defined by \(\varphi(x): t \mapsto \sin (x(t))\) is everywhere G芒teaux but nowhere Fr茅chet differentiable.

Short Answer

Expert verified
The operator \( \varphi(x) = \sin(x(t)) \) is G芒teaux differentiable in every direction but not Fr茅chet differentiable due to the nonlinearity of sine function.

Step by step solution

01

Understanding the operators

First, recall that a G芒teaux differentiable operator is one that has a directional derivative in every direction, while a Fr茅chet differentiable operator is one that has a linear approximation. A function can be G芒teaux differentiable without being Fr茅chet differentiable if it does not have a uniform linear approximation.
02

Define the Operator and Set up Requirements

Given the operator \( \varphi(x): t \mapsto \sin (x(t)) \), we need to check if it is G芒teaux differentiable in every direction and if it is nowhere Fr茅chet differentiable. Begin by writing the definition of G芒teaux and Fr茅chet differentiability.
03

G芒teaux Differentiability

For G芒teaux differentiability, compute the directional derivative of \( \varphi \). For any function \( x \in L_{2}[0,1] \) and direction \( h \in L_{2}[0,1] \), evaluate the limit: \( \varphi_x'(h) = \lim_{\varepsilon \to 0} \frac{ \varphi(x + \varepsilon h)(t) - \varphi(x)(t) }{ \varepsilon } = \lim_{\varepsilon \to 0} \frac{ \sin(x(t) + \varepsilon h(t)) - \sin(x(t)) }{ \varepsilon } = h(t) \cos(x(t)) \). Since this limit exists for all directions \( h \in L_{2}[0,1] \), the operator is G芒teaux differentiable.
04

Non-existence of Fr茅chet Differentiability

For Fr茅chet differentiability, \( \varphi \) must have a linear operator \L\ that serves as a good approximation. To find such a linear approximation: \( \| \varphi(x+h) - \varphi(x) - Lh \|_2 = o(\| h \|_2) \). If we let \( Lh = h(t) \cos(x(t)) \, \varphi(x+h)(t) - \varphi(x)(t) - Lh = \sin(x(t) + h(t)) - \sin(x(t)) - h(t) \cos(x(t)) \, using \sin(A+B) = \sin A \cos B + \cos A \sin B \,, \sin(x + h) - \sin(x) - h \cos(x) = (because of non-linearity of h's derivative\). \Thus \( \varphi \) is not Fr茅chet differentiable.

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Key Concepts

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

G芒teaux Differentiability
Understanding the concept of G芒teaux differentiability is key in functional analysis. When we say an operator is G芒teaux differentiable, we mean it has a directional derivative in any direction within the normed vector space. To explain further, imagine a function that changes as you nudge it in various directions. If the rate of this change can be calculated consistently in any direction, then the function is G芒teaux differentiable.

Consider the operator \(\textbackslash\textbackslash)varphi(x): t \mapsto \sin(x(t)) \) from the exercise. To check if it's G芒teaux differentiable, we compute the directional derivative for any function \(\textbackslash\textbackslash)x \in L_{2}[0,1] \) and direction \(\textbackslash\textbackslash)h \in L_{2}[0,1] \). The derivative is given by the limit:

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Most popular questions from this chapter

Show that a finite lower semicontinuous convex function \(f\) that is defined on a whole Banach space must be continuous.

Let \(X\) be a Banach space and let \(f\) be a continuous convex function on \(X^{*}\) that is \(w^{*}\) -lower semicontinuous. Show that if \(f\) is Fr茅chet differentiable at \(x^{*} \in X^{*}\), then \(f^{\prime}\left(x^{*}\right) \in X\). Hint: The derivative, as a uniform limit of quotients in \(B_{X^{*}}\), is also \(w^{*}\) -lower semicontinuous. Then use its linearity to see that \(f^{\prime}\left(x^{*}\right)\) is a functional that is \(w^{*}\) -continuous on \(B_{X *}\) and apply Theorem \(4.44\).

Prove that every convex function \(f\) defined on an open interval \(I \subset \mathbf{R}\) is differentiable at all but (at most) countably many points of \(I\). Hint: Observe that \(d^{+} f(x)(1)=\lim _{t \rightarrow 0+} \frac{f(x+t)-f(x)}{t}\), the derivative of \(f\) at \(x\) from the right, is a nondecreasing function of \(x\). Prove then that, at any point where \(f\) fails to be differentiable, the monotone function \(x \rightarrow\) \(d^{+} f(x)(1)\) has a jump. Because there are not more than a countable number of jumps, the conclusion follows.

Show that every exposed point of a convex set is extreme and give an example of an extreme point that is not exposed.

Find an example of a G芒teaux differentiable norm on a Banach space that is not Fr茅chet differentiable at some points. Hint: Any equivalent renorming by a G芒teaux differentiable norm of \(\ell_{1}\) (Theorem 8.13) satisfies the requircment (Theorem 8.26).
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