Question

Let \(X, Y\) be Banach spaces. (i) Let \(T\) be a one-to-one map from \(X\) onto \(Y\). Show that if \(C\) is a convex subset of \(X\), then \(T(\operatorname{Ext}(C))=\operatorname{Ext}(T(C))\) In particular, if \(T\) is an isometry of \(X\) into \(Y\), then \(T\left(\operatorname{Ext}\left(B_{X}\right)\right)=\) \(\operatorname{Ext}\left(T\left(B_{X}\right)\right)\) (ii) Let \(T\) be a bounded linear operator from \(X\) into \(Y .\) Show that if \(C\) is \(w\) -compact in \(X\), then \(\operatorname{Ext}(T(C)) \subset T(\operatorname{Ext}(C))\)

Short answer

For a one-to-one map T, we have T(Ext(C)) = Ext(T(C)). For an isometry, T(Ext(B_X)) = Ext(T(B_X)). If T is a bounded linear operator and C is weakly compact, Ext(T(C)) ⊆ T(Ext(C)).

Step-by-step solution

Title - Understand Convexity and Extreme Points

A set is convex if, for any two points within the set, the line segment that connects these points is also within the set. The extreme points of a convex set are those points that cannot be written as a convex combination of other points within the set.

- Definitions for Mapping

Let T be a one-to-one map from Banach space X onto Banach space Y, and let C be a convex subset of X. Define the image of the convex subset C under T as T(C) = {T(x) : x ∈ C}. Also, define the extreme points of C as Ext(C).

- Show T(Ext(C)) = Ext(T(C))

Take any extreme point x in Ext(C). Since T is one-to-one, T(x) is in T(C). Suppose T(x) is not an extreme point in T(C). Then, there exist y, z from T(C) such that T(x) is a non-trivial convex combination of y and z. Since y, z ∈ T(C), there exist y', z' in C such that T(y') = y and T(z') = z. Because T is linear and one-to-one, x must also be a non-trivial convex combination of y' and z', contradicting that x is an extreme point. Thus, T(x) must be in Ext(T(C)).

- Special Case for Isometry

An isometry is a map that preserves distances. For an isometry T of X into Y, if T(B_X) = B_Y where B_X and B_Y are unit balls in X and Y respectively, then from Step 3, we can state that T(Ext(B_X)) = Ext(T(B_X)) = Ext(B_Y).

- Definitions for Weak Compactness

A set is weakly compact if every sequence within that set has a subsequence that converges weakly within the set. Let T be a bounded linear operator from X into Y, and let C be a weakly compact subset of X.

- Show Ext(T(C)) ⊆ T(Ext(C))

Since T is linear and bounded, mapping extreme points of a weakly compact convex set remains within T(C). Given a point in Ext(T(C)), it must correspond to an extreme point in C because any extreme point in T(C) remains an extreme point when viewed through T. Therefore, Ext(T(C)) is a subset of T(Ext(C)).

Key concepts

Extreme Points

In convex sets, an extreme point is uniquely significant. Imagine the set as a shape, and the extreme points are the 'corners' or the furthest points that cannot be formed by mixing other points in the set.
They are crucial in various fields like optimization and functional analysis, notably because they help in characterizing and studying convex sets in depth.
To better understand, think of a triangle. The vertices are the extreme points because you cannot get these points by mixing any other points within the triangle.

• Convex Set: A set where any two points can be connected with a line segment that stays within the set.
• Extreme Points: Corner points of a convex set that can't be written as a blend of other points in the set.

One-to-One Map

A one-to-one map (or injective map) ensures that each element in the domain (X) maps to a unique element in the codomain (Y). This uniqueness is critical in showing relationships between extreme points.
When we talk about a one-to-one map in Banach spaces, it means no two different points in X map to the same point in Y. This property is central to proving that the image of extreme points under a one-to-one map remains preserved.

• Injective: Each element of X has a unique image in Y.
• Maintains uniqueness: Important for the structure of convex and extreme points.

Isometry

An isometry is a special kind of map that keeps the distance between points unchanged. In Banach spaces, it means transforming X to Y without distorting distances. This leads to the preservation of the geometric structure.
For example, if we have a triangle and we apply an isometric transformation like rotation or translation, the distances between any two vertices are unchanged. Hence, it naturally maintains the extreme points.

• Distance-preserving: Keeps original distances intact.
• Geometric fidelity: Ensures that extreme points remain as extreme points post-mapping.

Weak Compactness

Weak compactness refers to a set where every sequence has a weakly convergent subsequence within the set. It's like having an endless supply of subsequences that don't stray far.
In Banach spaces, weak compactness helps manage infinite-dimensional spaces by focusing on sequences and their limits. This property is handy when dealing with functional analyses involving operators and their continuity.

• Weak convergence: Sequences converge, but in a weaker sense.
• Subsequential limit: A sequence plus its converging subsequence.
• Maintains compact properties: Useful in infinite-dimensional spaces.

Bounded Linear Operator

A bounded linear operator maps elements between Banach spaces while controlling their 'size' or norm. Think of it as a function that scales inputs without blowing them out of proportion.
For example, multiplying by a constant is a bounded linear operator. When dealing with weakly compact sets, bounded linear operators help keep sets compact under the mapping.
This plays a role in preserving the structure and properties of sets when transforming from one Banach space to another.

• Linear: Maps straight lines to straight lines, preserving addition and scalar multiplication.
• Bounded: Keeps the norm (size) in control.
• Compactness preservation: Helpful in functional analysis.