Question

Let \(X, Y\) be Banach spaces, and let \(T\) be an isometric mapping of \(X\) onto \(Y\). The Mazur-Ulam theorem claims that if \(T(0)=0\), then \(T\) is necessarily linear. Show this result for the case when \(Y\) is strictly convex.

Short answer

Since T is isometric and Y is strictly convex, T must preserve linear structure due to distance preservation and strict convexity properties, proving T is linear.

Step-by-step solution

Understand Isometric Mapping

An isometric map preserves distances. For a map T: X → Y, this means that for all x1, x2 ∈ X, the equality \(\|T(x1) - T(x2)\|_Y = \|x1 - x2\|_X\) holds.

Use the Property T(0)=0

Since T(0) = 0, the mapping keeps 0 fixed. This is essential since it suggests T(b) = b for some b in Y when we compare image distances.

Apply the Mazur-Ulam Theorem

The Mazur-Ulam theorem states that if T is an isometric mapping that fixes 0, then T is linear. We need to show that this applies in the strictly convex case.

Consider the Strict Convexity of Y

In a strictly convex space, the midpoint of two distinct points lies strictly inside the line segment joining these points. Formally, if \(||z_1|| = ||z_2|| = 1\), and \(z_1 eq z_2 \), then \(||\frac{z_1+z_2}{2}|| < 1\).

Prove T is Linear

To prove linearity, consider two points x, y in X. By the properties of isometric maps and strict convexity: \[\begin{aligned} \|T(x + y)\|_Y & = \|x + y\|_X , \ \|T(x) + T(y)\|_Y & \leq \|T(x)\|_Y + \|T(y)\|_Y = \|x\|_X + \|y\|_X = \|x + y\|_X. \end{aligned}\] By strict convexity, \(\|T(x) + T(y)\|_Y = \|T(x + y)\|_Y\) must hold. This implies \(T(x + y) = T(x) + T(y)\).

Verify Homogeneity

Lastly, for any scalar \(\alpha\) and vector \(x\), note that \[ T(\alpha x) = \alpha T(x) \] using similar distance-preserving logic and strict convexity, confirming both properties of linearity.

Key concepts

Understanding Banach Spaces

A Banach space is a vector space equipped with a norm, which is complete concerning this norm. This means that any Cauchy sequence in the space will converge to a point within the same space. Examples of Banach spaces include spaces like \(L^p\) spaces for \(p \geq 1 \), where functions are defined with a specific integrable power.
Banach spaces are central in functional analysis because their completeness property ensures that the limit of a sequence of vectors behaves well under various operations.
This stability is crucial when dealing with mappings and transformations, ensuring that the results hold within the space.

• Vector space: A collection of vectors where addition and scalar multiplication are defined.
• Norm: A function assigning a non-negative length or size to each vector.
• Cauchy sequence: A sequence where the elements get closer to each other as the sequence progresses.
• Completeness: Every Cauchy sequence converges within the space.

Understanding Banach spaces lays the foundation for grasping more complex theorems, like the Mazur-Ulam theorem, that involve mappings between these spaces.

Isometric Mapping Explained

An isometric mapping is a function that preserves distances. This means if you measure the distance between two points in the source space and map those points to a target space, the distance between the images of these points remains unchanged.
Formally, for an isometric map \( T: X \rightarrow Y \), we have: \( \|T(x_1) - T(x_2)\|_Y = \|x_1 - x_2\|_X \) for all \( x_1, x_2 \in X \). This equality indicates that the structure or 'shape' of the space is preserved under the mapping.
An essential aspect of isometric mappings is that they are not only distance-preserving but also injective (one-to-one), meaning every point in the source space has a unique corresponding point in the target space.

• Preserving distances: The key property of isometric mappings.
• Injective: Each point in the source space maps to a unique point in the target space.
• Norm invariance: The norm of the difference between two points remains the same when mapped.

Isometric mapping is a crucial concept in understanding the Mazur-Ulam theorem, which states that such mappings are linear under specific conditions.

Strict Convexity in Spaces

A space is strictly convex if, for any two distinct points within a unit ball, the midpoint of the line segment joining these points lies strictly inside the unit ball. Mathematically, for \( \|z_1\| = \|z_2\| = 1 \) and \( z_1 \e z_2 \), we have \( \|\frac{z_1 + z_2}{2}\| < 1 \).
This property ensures that the space does not have any 'flat' parts on its boundary and encourages uniqueness in the configurations. In other words, combining points always results in points that do not stay on the boundary but lie strictly inside.

• Midpoint property: The midpoint of any two distinct points lies inside the segment.
• Uniqueness: Helps in ensuring properties like the linearity of isometric maps.

Strict convexity plays a vital role in the Mazur-Ulam theorem. It helps in proving that an isometric mapping, under the condition \( T(0) = 0 \), must be linear.