Question

Show that if \(X\) is a finite-dimensional Banach space, then every linear functional \(f\) on \(X\) is continuous on \(X\).

Short answer

Every linear functional on a finite-dimensional Banach space is continuous because linear functionals on \( \text{\bf R}^n \) are continuous and every finite-dimensional Banach space is isomorphic to \( \text{\bf R}^n \).

Step-by-step solution

- Understand the Given Problem

We need to show that any linear functional on a finite-dimensional Banach space is continuous. Let's begin by understanding the terms: A Banach space is a complete normed vector space, and a linear functional is a linear map from the space to the real or complex numbers.

- Finite-dimensional Spaces are Isomorphic to \(\text{\bf R}^n\)

Recall that any finite-dimensional Banach space is isomorphic to \(\text{\bf R}^n\) (or \(\text{\bf C}^n\) if working with complex numbers). An isomorphism here means there exists a linear bijection between the Banach space and \(\text{\bf R}^n\).

- Linear Maps on \( \text{\bf R}^n \) are Continuous

In finite-dimensional spaces such as \( \text{\bf R}^n \), any linear map is continuous. This is a known result from linear algebra and is based on the fact that all norms on finite-dimensional vector spaces are equivalent.

- Use the Isomorphism to Finalize the Proof

Given the isomorphism between our Banach space \(X\) and \( \text{\bf R}^n \), any linear functional on \(X\) can be translated into a linear functional on \(\text{\bf R}^n\). Since linear functionals on \( \text{\bf R}^n \) are continuous, so are the linear functionals on \(X\) due to the isomorphism.

Key concepts

linear functional

A linear functional is a very special kind of function in mathematics. It is a linear map that takes a vector from a vector space and returns a real or complex number. Importantly, it preserves the operations of addition and scalar multiplication. This means that for a linear functional \(f \) and any two vectors \(x, y \) in the vector space, and any scalar \(a, b \) from the respective field: \ f(ax + by) = af(x) + bf(y) \. This property makes linear functionals particularly useful in various fields including functional analysis, quantum mechanics, and optimization.
In simpler terms, a linear functional stretches or scales vectors and sums their results in a linear way, turning them into simple numbers. Understanding this will make further steps, such as studying continuity and isomorphism, clearer and more logical.

continuity

Continuity, in the context of functions, means that small changes in the input lead to small changes in the output. For linear functionals on finite-dimensional Banach spaces, continuity is guaranteed.
To see why, let's start by recalling that a Banach space is a vector space with a norm, which is complete (meaning it contains all its limit points). Now, in any real-world application, we love Banach spaces as they allow us to handle infinite sequences nicely under the operation of taking limits.
We can show that a linear functional \(f \) is continuous by noting that in a finite-dimensional Banach space, any linear map is continuous. This comes from the equivalence of all norms in finite-dimensional spaces. In other words, no matter which norm we use, distances and sizes of vectors remain proportional. Therefore, any small change in the vector results in a proportionally small change in the output number given by the linear functional, ensuring continuity.

isomorphism

Isomorphism is a concept that gives a really powerful idea: two mathematical structures can be considered the same if there is a way to map one structure onto the other perfectly, respecting their operations.
When we say a finite-dimensional Banach space is isomorphic to \( \text{\bf R}^n \), we mean there's a bijective (one-to-one and onto) linear map that preserves vector addition and scalar multiplication. Essentially, it means these spaces are structurally identical for practical purposes.
This is a critical concept because it tells us that what happens in the relatively simple and intuitive \( \text{\bf R}^n \) also happens in any finite-dimensional Banach space. Thus, properties like continuity of linear maps that are well-known in \( \text{\bf R}^n \) transfer directly. Therefore, if every linear functional is continuous on \( \text{\bf R}^n \), they are also continuous on any finite-dimensional Banach space through the isomorphism. Understanding isomorphisms helps in transferring and applying results from simpler spaces to more complex ones seamlessly.