Question

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

Short answer

The norm \(\backslash|\|x\|\| = \backslashsum_{i=1}^{\infty} \frac{\|x_i\|}{\log(i+1)}\) on \(\backslashell_{1}\) is Gâteaux differentiable but not Fréchet differentiable at some points.

Step-by-step solution

Understand the Concepts

Recognize the definitions of Gâteaux differentiability and Fréchet differentiability. A norm is Gâteaux differentiable at a point if the directional derivative exists in every direction, while it is Fréchet differentiable if this derivative is also continuous.

Identify Suitable Spaces

Consider the space \(\backslashell_{1}\), which can be renormed by a Gâteaux differentiable norm but is known not to be Fréchet differentiable everywhere (as per Theorems 8.13 and 8.26 from the hint).

Choose an Equivalent Norm

Select an equivalent norm for \(\backslashell_{1}\) that is Gâteaux differentiable. One such example is given by \(\backslash|\|x\|\| = \backslashsum_{i=1}^{\infty} \frac{\|x_i\|}{\log(i+1)}\).

Verify Gâteaux Differentiability

Show that the chosen norm \(\backslash|\|\| \ \backslashsum_{i=1}^{\infty} \frac{\|x_i\|}{\log(i+1)}\) is Gâteaux differentiable. This can be done by demonstrating the existence of the directional derivative.

Non-Fréchet Differentiability

Demonstrate that the chosen norm is not Fréchet differentiable at certain points. This involves showing that the directional derivative is not continuous.

Key concepts

Gâteaux Differentiability

The concept of Gâteaux differentiability relates to directional derivatives in functional analysis. A norm \( \| \cdot \| \) on a Banach space is said to be Gâteaux differentiable at a point \( x \) if the directional derivative exists in every direction. Formally, for any direction \( h \), the limit exists: \[ \lim_{t \to 0} \frac{\| x + th \| - \| x \|}{t}. \] This means that we can approximate how the norm changes in any given direction around the point \( x \).

Let's break this down:

• **Directional derivative**: Measures the rate of change of the norm in a specific direction.
• **Existence in all directions**: Must hold for every possible direction \( h \).

Keep in mind that Gâteaux differentiability is a less stringent requirement compared to Fréchet differentiability, as it only focuses on existence, not the continuity of the derivative.

Fréchet Differentiability

Fréchet differentiability is a stronger concept compared to Gâteaux differentiability. For a norm \( \| \cdot \| \) on a Banach space to be Fréchet differentiable at a point \( x \), the directional derivative must not only exist but also be continuous. This means that the derivative provides a linear approximation to the norm function around \( x \) uniformly in all directions. Formally:

There exists a bounded linear operator \( A \) such that: \[ \lim_{\| h \| \to 0} \frac{\| x + h \| - \| x \| - \langle A, h \rangle }{ \| h \|} = 0, \] where \( \langle A, h \rangle \) represents the action of the linear operator on the direction \( h \).

Here's what this means:

• **Linear approximation**: The norm function can be closely approximated by a linear map near \( x \).
• **Uniform continuity**: This approximation holds uniformly as \( h \) approaches zero.

Fréchet differentiability implies Gâteaux differentiability, but not vice versa. This means every Fréchet differentiable norm is also Gâteaux differentiable, but a Gâteaux differentiable norm might not be Fréchet differentiable.

Banach Space

A Banach space is a complete normed vector space. This means it has two main properties:

• **Normed**: There is a function (norm) \( \| \cdot \| \) that assigns a non-negative length or size to each vector in the space.
• **Complete**: Every Cauchy sequence in the space converges to a limit within the space.

Let's consider the space \( \ell_1 \), which consists of all infinite sequences of real (or complex) numbers whose series is absolutely convergent. The norm in this space is given by: \[ \| x \|_{\ell_1} = \sum_{i=1}^{\infty} |x_i|. \] \( \ell_1 \) is a Banach space under this norm.

Now, when we talk about *equivalent norms*, we mean different norms that define the same topology on the space. For instance, we might consider a new norm: \[ \| x \|_2 = \sum_{i=1}^{\infty} \frac{|x_i|}{ \log(i+1)}. \] It’s equivalent to the original \( \ell_1 \) norm but exhibits different differentiability properties.

To summarize: Banach spaces are foundational in functional analysis. Equivalent norms can change their properties (like differentiability) while retaining the key topological structure.