Question

Show that every norming subspace of \(X^{*}\) is a separating set for \(X\). Find an example of a separating subspace that is not norming.

Short answer

Norming subspaces are always separating. A subspace of \( \mathbb{R}^2 \) consisting of functionals \( f(x, y) = x \) is separating but not norming.

Step-by-step solution

Define Norming Subspace

A norming subspace of a dual space \( X^{*} \) is a subspace \( S \) such that for every \( x \) in \( X \), \( ||x|| = \sup_{f\in S, ||f||\leq1}|f(x)| \). This ensures the norm of \( x \) can be recovered precisely from functionals in \( S \).

Define Separating Set

A set \( S \) is called separating for \( X \) if for every \( x \) in \( X \) (excluding the zero vector), there exists \( f \) in \( S \) such that \( f(x) eq 0 \).

Prove Norming Implies Separating

Suppose \( S \) is a norming subspace. Take an \( x \) in \( X \) such that \( x eq 0 \). By the definition of \(||x|| \), there exists \( f \) in \( S \) such that \(|f(x)| = ||x|| \). Since \( ||x|| eq 0 \), \( f(x) eq 0 \). This shows \( S \) is separating as there is always some \( f \in S \) which does not annihilate \( x \).

Example of Separating but Not Norming Subspace

Consider \( X = \mathbb{R}^{2} \) and let \( S \) be the subspace of \( X^{*} \) consisting of linear functionals of the form \( f(x, y) = x \). This subspace is separating because for any \( (x, y) eq (0, 0) \) in \( \mathbb{R}^2 \), we can always find \( f \in S \) such that \( f(x, y) = x eq 0 \). However, this subspace is not norming because \(||(0, 1)|| = 1 \) but \( \sup_{f \in S, ||f|| \leq 1}|f(0, 1)| = 0 \).

Key concepts

norming subspace

In functional analysis, a norming subspace plays a critical role in understanding the structure and properties of a given vector space. Let's dive deeper into this concept. A subspace \( S \) of the dual space \( X^{*} \) is called norming if for every element \( x \) in the original space \( X \), the norm of \( x \) can be determined using the functionals in \( S \). Formally, this is expressed as:\( ||x|| = \sup_{f \in S, ||f|| \leq 1}|f(x)| \). This means that the norm of any vector \( x \) in \( X \) can be completely described by the supremum (or least upper bound) of the absolute values of the functionals in \( S \) acting on \( x \). This relationship ensures that \( S \) captures enough information about \( X \) to

separating set

A separating set is essential for distinguishing between different elements of a vector space. A set \( S \) is defined to be separating for a space \( X \) if for every non-zero element \( x \) in \( X \), there exists a functional \( f \) in \( S \) such that \( f(x) \) is not equal to zero. In simpler terms, a separating set can always 'detect' different vectors. This concept is very useful in analysis, as it ensures that any vector in \( X \) can be 'separated' or identified uniquely by some functional in \( S \). This is crucial for many theorems and applications in functional analysis, where distinguishing between vectors is essential.

dual space

The dual space \( X^{*} \) of a vector space \( X \) consists of all linear functionals from \( X \) to the underlying field. Linear functionals are mappings \( f: X \rightarrow \mathbb{K} \) (usually \( \mathbb{R} \) or \( \mathbb{C} \)) that satisfy linearity, which means \( f(ax + by) = af(x) + bf(y) \) for all scalars \( a, b \) and vectors \( x, y \) in \( X \). The dual space is itself a vector space, where addition and scalar multiplication are defined pointwise. The study of dual spaces is important because they provide a way to 'test' or 'probe' the original space using functionals. Many properties of the vector space \( X \) can be analyzed using its dual space, making it a powerful tool in functional analysis.

linear functionals

Linear functionals are at the heart of functional analysis. A linear functional \( f \) on a vector space \( X \) is a function that takes a vector \( x \) in \( X \) and returns a scalar from the underlying field (such as real or complex numbers). Linear functionals must satisfy two properties: linearity and homogeneity. This means, for any vectors \( x \) and \( y \) in \( X \) and scalars \( \alpha \) and \( \beta \): \( f(\alpha x + \beta y) = \alpha f(x) + \beta f(y) \). These functionals help in understanding and characterizing the space \( X \) by providing scalar values that represent different aspects of the vectors in \( X \). They form the elements of the dual space \( X^{*} \), hence playing a significant role in various key concepts such as norming subspaces and separating sets.