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Show that the norm of \(C[0,1]\) is nowhere Fr茅chet differentiable. Show that the norm of \(C[0,1]\) is G芒teaux differentiable at \(x \in S_{C[0,1]}\) if and only if \(|x|\) attains its maximum at exactly one point of \([0,1]\). Hint: Note that the distance between two different Dirac measures in \(C[0,1]^{*}\) is two. Given \(x \in S_{C[0,1]}\), choose \(t_{0} \in[0,1]\) such that \(x\left(t_{0}\right)=1\). Then choose \(t_{n} \neq t_{0}\) such that \(x\left(t_{n}\right) \rightarrow 1\). By the 艩mulian lemma, \(x\) is not a point of Fr茅chet differentiability of the supremum norm on \(C[0,1]\). For the second part, assume that \(x \in S_{C[0,1]}\) is such that \(x\left(t_{0}\right)=1\) and \(|x(t)|<1\) for every \(t \neq t_{0} .\) Put \(H=\left\\{f \in C[0,1]^{*} ;\|f\| \leq 1, f(x)=1\right\\} .\) If \(H \cap B_{C[0,1] *} \neq\left\\{\delta_{t_{0}}\right\\}\), then this intersection would have at least two extreme points that would be extreme points of \(B_{C[0,1]^{*}}\). All the extreme points of \(B_{C[0,1]^{*}}\) are \(\pm\) Dirac measures (Lemma 3.42).

Step by step solution

01

- Understanding the Norm on \(C[0,1]\)

The norm on \(C[0,1]\) is the supremum norm, defined as \(orm{x} = \text{sup}_{t \ in [0,1]} |x(t)|\). This means that the norm of a function is the maximum absolute value it attains on the interval \([0,1]\).

02

- Fr茅chet Differentiability

A function is Fr茅chet differentiable at a point if the limit:\[\frac{orm{f + h} - orm{f} - L(h)}{orm{h}} \rightarrow 0 \text{ as }orm{h} \rightarrow 0\]exists, where \(L\) is a linear function. To show the supremum norm on \(C[0,1]\) is not Fr茅chet differentiable anywhere, we will use a contradiction with the 艩mulian lemma.

03

- 艩mulian Lemma

According to the 艩mulian lemma, a function \(x \ in S_{C[0,1]}\) is not a point of Fr茅chet differentiability if there exists a sequence \(t_n \ in [0,1]\) distinct from \(t_0\) such that \(x(t_n) \rightarrow x(t_0)\). Since for any chosen \(t_0 \ in [0,1]\), one can find such a sequence, \(x\) is not a point of Fr茅chet differentiability.

04

- G芒teaux Differentiability

A function \(x \ in S_{C[0,1]}\) is G芒teaux differentiable at \(x\) if the directional derivative exists in every direction. If \(orm{x}\) attains its maximum at exactly one point \(t_0\), then \(x\) is G芒teaux differentiable at \(x\) because the supremum norm gradient exists uniquely.

05

- Uniqueness of Attainment Point

Assume \(x \ in S_{C[0,1]}\) such that \(x(t_0) = 1\) and \(|x(t)| < 1\) for every other \(t\). Consider the set \(H = \{ f \ in C[0,1]^* ; \orm{f} \leq 1, f(x) = 1 \} \). If \(H \cap B_{C[0,1]^*} \eq \{ \delta_{t_0} \}\), then \(H\) would contain at least two extreme points, which would result in contradictions with extreme point definitions.

06

- Dirac Measures

All extreme points of \(B_{C[0,1]^*}\) are \(\pm\) Dirac measures. If \(x(t_n) \rightarrow 1\) for a sequence \(t_n \eq t_0\), there are at least two extreme points of \(B_{C[0,1]^*}\), leading to a contradiction. Hence, the supremum norm is G芒teaux differentiable at \(x \in S_{C[0,1]}\) if and only if \(orm{x}\) attains its maximum at exactly one point \([0,1]\).

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Key Concepts

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

Fr茅chet differentiability

In functional analysis, Fr茅chet differentiability is a concept that extends the usual notion of differentiation from calculus to more complex spaces. A function is Fr茅chet differentiable at a point if there exists a linear function (called the differential) that approximates the change in the function at that point. Mathematically, a function \(f\) is Fr茅chet differentiable at \(x\) if the limit:
\[ \lim_{{\|h\| \to 0}} \frac{\|f(x+h) - f(x) - L(h)\|}{\|h\|} = 0 \]
exists, where \(L\) is a linear map. This means that the error in the linear approximation by \(L\) becomes negligible as \(h\) approaches zero. In the context of the supremum norm on \(C[0,1]\), it鈥檚 important to recognize that this strong form of differentiability is not generally exhibited, as highlighted by the 艩mulian lemma. The lemma helps in determining the non-differentiability through readily available discrete sequences.

G芒teaux differentiability

G芒teaux differentiability is a weaker form of differentiation than Fr茅chet differentiability. A function is G芒teaux differentiable if the directional derivative exists in every direction. Specifically, for a function \(f\) at point \(x\), the G芒teaux derivative in the direction \(h\) is given by:
\[ Df(x;h) = \lim_{t \to 0} \frac{f(x + th) - f(x)}{t} \]
Unlike Fr茅chet differentiability, this does not require the existence of a linear map that approximates the change in all directions simultaneously. In case the supremum norm in \(C[0,1]\), we observe that G芒teaux differentiability at a point \(x\) depends on the function achieving its maximum at exactly one point. This distinct maximum ensures that the directionality criterion necessary for G芒teaux differentiability is satisfied.

艩mulian lemma

The 艩mulian lemma is a crucial tool in functional analysis. It states that a function in a normed space \(S_{C[0,1]}\) is not a Fr茅chet differentiable point of the norm if there exists a sequence \(t_n\) such that \(t_n eq t_0\) and \(x(t_n) \rightarrow x(t_0)\). Essentially, this lemma provides a criterion to establish non-differentiability by examining the function鈥檚 behavior at discrete points. When applying this lemma to the supremum norm, it implies that continuous functions achieving similar values in close yet distinct domains fail to meet the strict criteria of Fr茅chet differentiability.

Supremum norm

The supremum norm, also known as the infinity norm or the \(L^{\infty}\) norm, measures the maximum absolute value a function attains over a specified interval. For a continuous function \(x\) on \([0,1]\), the supremum norm is defined as:
\[ \|x\| = \sup_{t \in [0,1]} |x(t)| \]
This norm is particularly useful in spaces of bounded functions, such as \(C[0,1]\). It is less sensitive to small oscillations compared to other norms like the \(L^2\) norm, but it offers a clear bound on the function鈥檚 magnitude. Because of these properties, the supremum norm is often used in functional spaces to gauge the 鈥渟ize鈥� of functions, particularly in contexts that require strict control over their behavior across an interval.

Dirac measure

Dirac measures are a form of distribution used in functional analysis and probability theory. The Dirac measure centered at a point \(t_0\), denoted \(\delta_{t_0}\), assigns the entirety of its 鈥渨eight鈥� to the single point \(t_0\). For any measurable set \(A\), the Dirac measure is defined as:
\[ \delta_{t_0}(A) = \begin{cases} 1 & \text{if} \ t_0 \in A \ 0 & \text{if} \ t_0 otin A \end{cases} \]
In the context of functional spaces such as \(C[0,1]^*\), Dirac measures are critical in understanding extremal properties and differentiability. These measures are used to pinpoint positions where functions achieve extreme values and are particularly helpful when dealing with linear functionals. They also play a part in determining extreme points of certain sets, leading to insights into the geometric properties of these spaces.