Question

Show that \(c_{0}\) contains no two-dimensional subspace on which the standard norm is Gâteaux differentiable. Hint: Then some quotient \(Q\) of \(\ell_{1}\) would have uncountably many extreme points since the dual to a two-dimensional smooth space is strictly convex. Every point of the sphere of \(Q\) that is identified with the restriction to the two-dimensional subspace in question extends to an extremal point of the sphere in \(\ell_{1}\) by the Krein-Milman theorem, considering the face of all the extensions. Thus, there are uncountably many such extreme points of the ball of the standard norm in \(\ell_{1}\), which is a contradiction because the extreme points of the ball of \(\ell_{1}\) are exactly \(\pm e_{i}\).

Short answer

No two-dimensional subspace of \(c_0\) makes the standard norm Gâteaux differentiable because it would lead to uncountably many extreme points in \(\ell_1\), which is impossible.

Step-by-step solution

Understand the Problem

The problem is to show that the space of sequences converging to zero, denoted as \(c_0\), does not contain any two-dimensional subspace where the standard norm is Gâteaux differentiable.

Analyze the Hint

The hint tells us that if \(c_0\) had such a two-dimensional subspace, some quotient \(Q\) of \(\ell_1\) would have uncountably many extreme points. We need to show that this leads to a contradiction to prove the result.

Recall the Krein-Milman Theorem

The Krein-Milman theorem states that a convex compact set in a topological vector space is the closed convex hull of its extreme points. We'll use this to show the contradiction.

Relate to the Dual Space

The dual space to a two-dimensional smooth space is strictly convex. This means any point on the sphere of \(Q\) extends to an extremal point in \(\ell_1\).

Count the Extreme Points

For the space \(\ell_1\), the extreme points of the unit ball are exactly \(\pm e_i\), where \(e_i\) are the unit vectors. These are countable.

Reach a Contradiction

If there were uncountably many extreme points in the quotient space \(Q\), this would contradict the fact that \(\ell_1\) has only countably many extreme points. Therefore, no such two-dimensional subspace exists in \(c_0\).

Key concepts

Two-Dimensional Subspaces

A subspace is a space that is fully contained within a larger space and follows all the rules of that space. When we talk about two-dimensional subspaces, we mean spaces that can be described with just two dimensions.
In simpler terms, think of a flat sheet of paper within a 3D room. The room represents the larger space, and the flat sheet represents a two-dimensional subspace.
For the context of this problem, we are interested in whether such a two-dimensional subspace within the space of sequences that converge to zero (\(c_{0}\)) can have a Gâteaux differentiable norm. This means we want to check if the 'flat sheet' within our more complex space allows for certain mathematical properties connected to how functions change within that sheet.

The Space c₀

The space denoted as \(c_{0}\) consists of all sequences of real numbers that converge to zero. Each sequence in this space can be thought of as an infinite list of numbers that gets closer and closer to zero as you go further out in the list.
Elements of \(c_{0}\) are sequences like (1, 1/2, 1/3, ..., 1/n, 0, 0, 0, ...). These sequences don't need to get to zero right away, but they must tend toward zero.
This space is interesting because it's structured so that the sequences are controlled in a way that we can make broad, general statements about their properties and the spaces they contain. The problem we're tackling is to determine if within this space, there's a two-dimensional subspace that behaves in a specific, smooth way when considering norms and derivatives.

Krein-Milman Theorem

The Krein-Milman theorem plays a key role in this problem. This theorem states that any convex compact set in a topological vector space can be entirely described by its extreme points. Let's break that down:

• Convex: A set is convex if you can draw a straight line between any two points in the set, and that line stays inside the set.
• Compact: A set is compact if it is closed (contains all its boundary points) and bounded (doesn’t go off to infinity).
• Extreme Points: These are points in the set that cannot be written as a combination of two other points in the set. They're the 'corners' of the set.

In our context, the Krein-Milman theorem helps us argue about the structure of our space and its subspaces. By showing contradictions based on the number of extreme points, we help prove that no two-dimensional subspace in \(c_{0}\) can have the Gâteaux differentiable property for the norm.

ℓ₁ Space

The \(ℓ_{1}\) space is another space of sequences, similar to \(c_{0}\), but with a different condition. An element in \(ℓ_{1}\) is a sequence whose series is absolutely summable.
This means for any sequence (a₁, a₂, a₃, ...), the sum |a₁| + |a₂| + |a₃| + ... is finite. For example, the sequence (1, 1/2, 1/3, 1/4, ...) belongs to \(ℓ_{1}\) because, despite having infinitely many terms, the sum of their absolute values is finite.
In the context of this problem, the \(ℓ_{1}\) space's extreme points are the sequences where one element is ±1 and all others are 0, like (1, 0, 0, ...) or (0, -1, 0, ...). These points are crucial for our contradiction argument because they help show that the number of extreme points in any quotient space derived from \(ℓ_{1}\) must be countable, not uncountable.

Strict Convexity

Strict convexity is a property of spaces that makes their structure 'rounder' in a precise mathematical sense. In a strictly convex space, any line segment connecting two points in the space lies entirely within the interior of the space, except for the endpoints.
Practically, think of a circle, where every line segment between two points on the circle lies entirely inside the circle except for the very ends.
For the problem at hand, when we talk about the dual space to a two-dimensional smooth space being strictly convex, we're leveraging the fact that this smooth space has a nice, round, well-behaved structure. This helps in proving that certain subspaces cannot exist within \(c_{0}\) with the desired differentiability properties.
It boils down to showing that the smoothness and roundness principles in strictly convex spaces contradict the mathematical properties we'd expect if such a two-dimensional subspace existed in \(c_{0}\). Hence, leading to our conclusion.