Question

Let \(\left\\{x_{i}\right\\}_{i=1}^{n}\) be a linearly independent set of vectors in a Banach space \(X\) and \(\left\\{\alpha_{i}\right\\}_{i=1}^{n}\) be a finite set of real numbers. Show that there is \(f \in X^{*}\) such that \(f\left(x_{i}\right)=\alpha_{i}\) for \(i=1, \ldots, n\). Hint: Define a linear functional \(f\) on \(\operatorname{span}\left\\{x_{i}\right\\}\) by \(f\left(x_{i}\right)=\alpha_{i}\) for \(i=1, \ldots, n\) and use the Hahn-Banach theorem.

Short answer

Extend the linear functional \(f\) using the Hahn-Banach theorem to obtain the desired \(\tilde{f} \in X^{*}\).

Step-by-step solution

Define a Linear Functional on the Span

Define the linear functional \(f\) on the span of \(\{x_{i}\}\) by setting \(f(x_{i}) = \alpha_{i}\) for \(i = 1, \ldots, n\). This means that for any vector in \(\operatorname{span}\{x_{i}\}\), we can express it as a finite linear combination \(\sum_{i=1}^n c_{i}x_{i}\) where \(c_{i}\) are scalars.

Verify Linearity of \(f\)

For any vectors \(u, v \in \operatorname{span}\{x_{i}\}\) and scalars \(a, b\), ensure that \(f(au + bv) = af(u) + bf(v)\) to confirm that \(f\) is a linear functional. Because \(u\) and \(v\) can be expressed as linear combinations of \(x_{i}\), it follows that \(f\) is linear by its definition on \(x_{i}\).

Apply the Hahn-Banach Theorem

Invoke the Hahn-Banach theorem, which allows the extension of the linear functional \(f\) defined on \(\operatorname{span}\{x_{i}\}\) to the entire Banach space \(X\). There exists an extension \(\tilde{f} \in X^{*}\) such that \(\tilde{f}(x_{i}) = f(x_{i}) = \alpha_{i}\) for \(i = 1, \ldots, n\) and \(\tilde{f}\) is a bounded linear functional.

Conclusion

Thus, there exists a functional \(\tilde{f} \in X^{*}\) such that \(\tilde{f}(x_{i}) = \alpha_{i}\) for \(i = 1, \ldots, n\), satisfying the condition required by the problem.

Key concepts

Banach spaces

A Banach space is a complete normed vector space. This means not only is it a vector space, but it also has a norm (a way to measure the length of vectors) and every Cauchy sequence in the space converges to a point within the space.

It's named after the mathematician Stefan Banach, who studied these spaces in the early 20th century. When we refer to a space being 'complete', we mean that it doesn't matter how many steps we take (as long as they get smaller and smaller), we'll always end up somewhere in the space. This is very useful in functional analysis and various practical applications.

Examples of Banach spaces include:

• The set of real numbers \(\mathbb{R}\)
• The set of continuous functions on a closed interval
• The set of square-integrable functions (integrals are finite)

Many function spaces studied in analysis are Banach spaces.

Hahn-Banach theorem

The Hahn-Banach theorem is a cornerstone of functional analysis. It allows us to extend the domain of a linear functional from a subspace to the entire Banach space without losing linearity or increasing its norm.

In simple terms, suppose we start with a linear functional defined on a smaller subset of the Banach space, the Hahn-Banach theorem ensures that we can find a linear functional over the whole space that behaves the same way on the smaller subset.

This is crucial because it shows us that linear functionals are very flexible and can be controlled well. They play a big role when dealing with spaces of functions, optimization problems, and differential equations.

Here's how we use the Hahn-Banach theorem in our exercise:

• First, we define a linear functional on a small part of our space (specifically, the span of our \(\{x_{i}\}\)).
• Then, we use the theorem to extend this functional to the whole Banach space, ensuring the properties we need are retained.

linear functional

A linear functional is a special kind of function that maps vectors to the underlying field (like real numbers) while respecting the operations of vector addition and scalar multiplication.

This means if you have two vectors \(u\) and \(v\) and a linear functional \(f\), then \(f(au + bv) = af(u) + bf(v)\) for any scalars \(a\) and \(b\). This linearity property makes them very useful in functional analysis.

Linear functionals are important because they help us understand the structure of vector spaces and their subspaces. They appear in many areas such as quantum mechanics, economics, and optimization. In the context of our exercise:

• We start with a finite set of vectors and define a linear functional that acts on their span.
• We then use the Hahn-Banach theorem to extend this to the whole Banach space, ensuring that our original conditions are satisfied.

Linear functionals connect deeply with concepts like dual spaces and inner product spaces, making them a fundamental idea in advanced mathematics.