Question

\(\mathbf{}\) Let \((X,\|\cdot\|)\) be a Banach space. Show that \(\|\cdot\|^{2}\) is Fréchet differentiable at 0 and \(\left(\|\cdot\|^{2}\right)^{\prime}(0)=0\) Let \((H,\|\cdot\|)\) be a Hilbert space \(H\). Show that \(\|\cdot\|^{2}\) is Fréchet differentiable at every point of \(H\). Hint: The first part follows by direct calculation. \(F(h)=2(x, h)\) in the notation of Definition \(8.1 .\)

Short answer

\(\|\cdot\|^2\) is Fréchet differentiable at 0 with derivative 0 in any Banach space, and it is Fréchet differentiable everywhere in Hilbert space.

Step-by-step solution

Understanding Fréchet Differentiability

To show that a function is Fréchet differentiable at a point, we need to check that the function can be approximated by a linear map plus a term that goes to zero faster than linearly.

Show \(\|\cdot\|^{2}\) is Fréchet differentiable at 0 in a Banach space

Consider \(\phi(x) = \|x\|^2\). We need to check if \(\phi\) is Fréchet differentiable at \(0\): \[\phi'(0) = \lim_{\|x\| \to 0} \frac{\|x\|^2 - \|0\|^2}{\|x\|} = \lim_{\|x\| \to 0} \|x\| = 0.\]The Fréchet derivative \phi'(0) = 0\.

Define and use the hint in the Hilbert space

Let \(H\) be a Hilbert space and consider \[\phi(x) = \|x\|^2.\]For any \(h \in H\), use the hint: Assume \(F(h) = 2(x, h)\).\[\lim_{\|h\| \to 0} \frac{\|x+h\|^2 - \|x\|^2 - 2(x, h)}{\|h\|} = \lim_{\|h\| \to 0} \frac{(x+h, x+h) - (x, x) - 2(x, h)}{\|h\|} = 0.\]

Conclusion

From the above, we conclude that \(\phi(x) = \|x\|^2\) is Fréchet differentiable for all \(x \in H\), and the Fréchet derivative at any point \(x\) is the linear map \(2(x, \cdot)\).

Key concepts

Banach Space

A Banach space is a type of vector space. It is characterized by its completeness, meaning every Cauchy sequence in this space converges within the space.
Banach spaces are equipped with a norm \(orm{\bullet}\) that measures the 'length' of vectors.
This norm satisfies three main properties:

• Positive definiteness: \(orm{x} \geq 0\) and \(orm{x} = 0\) if and only if \x = 0\
• Homogeneity: \(orm{\boldsymbol{\beta} x} = \boldsymbol{\beta} orm{x}\) for any scalar \(\beta\)
• Triangle inequality: \(orm{x + y} \leq orm{x} + orm{y}\)

Banach spaces are foundational in functional analysis and are used in various fields of mathematics.

Hilbert Space

A Hilbert space is another important type of vector space. Like Banach spaces, Hilbert spaces are also complete with respect to their norm.
However, a key difference is that Hilbert spaces have an inner product \((\bullet, \bullet)\) that defines the norm. The norm is given by \(orm{x} = \big(\boldsymbol{\beta}, \boldsymbol{\beta}\big)^{1/2}\).
Inner products allow the measurement of angles between vectors and facilitate concepts like orthogonality and projection.

• Example of Hilbert spaces: \(\boldsymbol{\beta}\big)^2\) (space of square-integrable functions) and \(\boldsymbol{\beta}\big)^{\text{2}} =(\boldsymbol{\beta}, \boldsymbol{\beta}\big)^{d}\) (n-dimensional Euclidean space)
• Hilbert space can represent an infinite-dimensional vector.

These spaces are crucial in quantum mechanics, signal processing, and various branches of mathematics.

Fréchet Differentiation

Fréchet differentiation extends the concept of differentiation from real-valued functions to functions between Banach spaces.
A function \(F: X \to Y\) is Fréchet differentiable at a point \(x \in X\) if there is a bounded linear operator \(A: X \to Y\) such that:

\[ F(x + h) = F(x) + A(h) + o(orm{h}) \]
Here, \( o(orm{h})\) represents a term that goes to zero faster than linearly as \( h \to 0\).

• This concept allows us to approximate functions between Banach spaces with linear maps.
• In the context of the given exercise, the function \( orm{x}^2\) is shown to be Fréchet differentiable in the spaces in question.

Norm Squared Differentiability

The norm squared function, \( orm{x}^2\), measures the 'length' of a vector squared.
Interestingly, in many contexts, this norm squared is easier to differentiate than the norm itself.
In a Banach space, it's shown that \( orm{x}^2 \) is Fréchet differentiable at zero:

\[ \boldsymbol{\beta}(0) = \boldsymbol{\beta}\big(\boldsymbol{\beta}, h\big)\]

• In this case, the derivative evaluates to zero because the norm of zero is zero.
• In a Hilbert space, the norm squared function is shown to be Fréchet differentiable at every point. The derivative at any point \ x \ is given by the linear map \ 2(x,h) \>.
This simplification provided by the squared norm is particularly useful in optimization and analysis.

Linear Maps

Linear maps, or linear operators, are functions between vector spaces that preserve the vector space operations (addition and scalar multiplication). For example:

• \( T(a_1 u_1 + a_2 u_2) = a_1 T(u_1) + a_2 T(u_2) \)

Linear maps play a crucial role in various areas such as differential equations, where they represent systems with linear behavior, and in functional analysis, where they help study spaces and transformations between them.
• A bounded linear operator (or continuous) is one that maps bounded sets to bounded sets, and is essential for Fréchet differentiability.
• In the given exercise, the linear map \(2 (x,h)\) arises as the Fréchet derivative in the Hilbert space case.