Question

Let \(X\) be a Banach space. Show that if the norm of \(X\) is LUR (in particular, if \(X\) is a Hilbert space), then every \(x \in S_{X}\) is a strongly exposed point of \(B_{X}\).

Short answer

Any point in the unit sphere of a Banach space with an LUR norm (such as a Hilbert space) is a strongly exposed point of the unit ball.

Step-by-step solution

Define Basic Terms

Identify what it means for a point to be strongly exposed and for a norm to be LUR (Locally Uniformly Rotund). A point \(x \in S_X\) is strongly exposed if there exists a functional \(f\in X^*\) such that for all sequences \((x_n)\subseteq B_X\) with \(f(x_n) \to f(x)\), it follows that \(||x_n - x|| \to 0\). A norm is LUR if for any sequence \((x_n)\subseteq S_X\) with \(||x_n + x|| \to 2\), it follows that \(||x_n - x|| \to 0\).

Consider the Definition of LUR Norm

Recall that for a norm to be LUR, if a sequence \((x_n)\subseteq S_X\) satisfies \(\|x_n + x\| \to 2\), then we must have \(\|x_n - x\| \to 0\). This property hints at a connection between closeness in norm and the evaluation by a linear functional.

Sequence Convergence Condition

Given that \(X\) has an LUR norm, we choose any \(x \in S_X\). Suppose \((x_n)\subseteq B_X\) is such that \(f(x_n) \to f(x)\) for some \(f \in X^*\) with \(f(x) = 1\). We need to show \(\|x_n - x\| \to 0\).

Use LUR Property

Since \(x\) lies in the unit sphere and \(f(x)=1\), and considering the functional achieves its norm on \(x\), for large \(n\), \(x_n\) must be close to the boundary of \(B_X\). Particularly, we have \(\|x_n\| \approx 1\).

Convergence in Norm

Using the fact that \(f(x_n) \to f(x)\) and \(\|x\| = 1 = f(x)\), if \(\|x_n + x\| \to 2\), the LUR property implies \(\|x_n - x\| \to 0\). This follows since if \(x_n\) were not converging to \(x\) in norm, the LUR property would be contradicted.

Conclusion

Therefore, \(x\) is strongly exposed by the functional \(f\), as for any sequence \((x_n)\) such that \(f(x_n) \to f(x)\), \(||x_n - x|| \to 0\). This completes the proof.

Key concepts

Banach Space

A Banach space is a specific type of vector space. It is defined as a complete normed vector space, meaning every Cauchy sequence in this space converges to a point within the same space. Understanding what makes a Banach space is crucial because the concept of completeness assures us that limits exist within the space for every converging sequence.
Here are some key points to remember:

• Every Banach space has a norm, a function that measures the 'length' of vectors.
• The space must be complete, meaning it includes all its limit points.
• Banach spaces generalize many classical function spaces, such as spaces of continuous functions or integrable functions.

Proper grasp of these principles helps in understanding why certain theorems, like the LUR norm property, hold in Banach spaces.

Strongly Exposed Point

A strongly exposed point is a special type of point within a Banach space. In simple terms, a point \(x\) in a Banach space is considered strongly exposed if there exists a linear functional that can isolate this point strongly in a specific way.

• Given a point \(x \) in the unit sphere \(S_X\), it is strongly exposed if there is a linear functional \( f \) such that for any sequence \( (x_n) \) in \( B_X \) where \( f(x_n) \) approaches \ f(x) \, we get \| x_n - x \. This usage of linear functionals adds a layer of structure to our understanding, helping to prove properties like LUR in Banach spaces.
• This isolation or 'exposure' of \ x \ leads to important implications in optimization and geometry of Banach spaces.

Hilbert Space

A Hilbert space is another critical concept in functional analysis and generalizes the notion of Euclidean spaces to potentially infinite-dimensional spaces.
A Hilbert space introduces the idea of an inner product, which allows for defining angles and orthogonality.
Here are some important aspects:

• A Hilbert space is a complete inner-product space. This means it has a norm derived from an inner product.
• Every Hilbert space is a Banach space, but not every Banach space is a Hilbert space since not all Banach spaces have an inner product.
• The geometry of Hilbert spaces often makes them easier to work with compared to general Banach spaces. For example, the Pythagorean theorem and orthogonal projections are directly applicable in Hilbert spaces, simplifying many problems.

Comprehending Hilbert spaces provides a foundation for understanding advanced topics in quantum mechanics and many areas of applied mathematics.

Linear Functional

A linear functional is a fundamental tool in functional analysis. Essentially, it's a linear map from a vector space to the field of scalars, typically the real or complex numbers.
Here's a deeper look:

• Linear functionals are essential for studying spaces of functions. They help in defining dual spaces and norms.
• For a given vector \( x \) in the space, applying a linear functional results in a scalar value. This scalar value can be understood as a measure or evaluation of the vector under the functional.
• In the context of Banach spaces, linear functionals are used to expose points strongly. Specifically, a functional \( f  \) applied to a sequence \(( x_n )\) gives insight into the proximity of this sequence to a strongly exposed point \( x\) by observing the convergence of \( f(x_n) \).

Understanding linear functionals opens up many avenues in functional analysis, including the famous Hahn-Banach theorem that allows extension of functionals in powerful ways.