Question

Let \(X, Y\) be Banach spaces and let \(b\) be a map from \(X \oplus Y\) into \(\mathbf{R}\). We say that \(b\) is a bilinear form on \(X \oplus Y\) if \(b\left(x, y_{0}\right)\) is linear in \(x\) for every \(y_{0} \in Y\) and \(b\left(x_{0}, y\right)\) is linear in \(y\) for every \(x_{0} \in X\). Consider such a bilinear form \(b\) on \(X \oplus Y\). Show that the following are equivalent: (i) \(b\) is continuous at the origin \((0,0)\) of \(X \oplus Y\). (ii) \(b\) is uniformly continuous on bounded sets of \(X \oplus Y\). (iii) There is \(K>0\) such that \(|b(x, y)| \leq K\|x\|\|y\|\) for all \(x \in X\) and \(y \in Y\). (iv) \(b\) is separately continuous on \(X \oplus Y\); that is, \(b\left(x, y_{0}\right)\) is continuous in \(x\) for every \(y_{0} \in Y\), and \(b\left(x_{0}, y\right)\) is continuous in \(y\) for every \(x_{0} \in X\).

Short answer

The conditions (i), (ii), (iii), and (iv) are equivalent.

Step-by-step solution

Define Bilinear Form

First, understand that a bilinear form on the direct sum space \(X \oplus Y\) is a map \(b\) such that \(b(x, y_0)\) is linear in \(x\) for every \(y_0 \in Y\) and \(b(x_0, y)\) is linear in \(y\) for every \(x_0 \in X\).

Show that (i) implies (ii)

If \(b\) is continuous at the origin \((0,0)\) of \(X \oplus Y\), then for any \(\epsilon > 0\), there exists \(\delta > 0\) such that if \(\|x\| < \delta\) and \(\|y\| < \delta\), then \(|b(x, y)| < \epsilon\). This implies that \(b\) is uniformly continuous on bounded sets of \(X \oplus Y\).

Show that (ii) implies (iii)

Uniform continuity on bounded sets implies that there exists a constant \(K > 0\) such that \(|b(x, y)| \leq K \|x\| \|y\|\) for all \(x \in X\) and \(y \in Y\). This is because on the bounded set given by the norm \(\|x\| = 1\) and \(\|y\| = 1\), the values of \(b\) must also be bounded.

Show that (iii) implies (iv)

Given \(|b(x, y)| \leq K \|x\| \|y\|\), fix any \(y_0 \in Y\); then \(|b(x, y_0)| \leq K \|x\| \|y_0\|\), which implies \(b(x, y_0)\) is continuous in \(x\). Similarly, fix any \(x_0 \in X\); then \(|b(x_0, y)| \leq K \|x_0\| \|y\|\), which implies \(b(x_0, y)\) is continuous in \(y\). Thus, \(b\) is separately continuous on \(X \oplus Y\).

Show that (iv) implies (i)

If \(b(x, y_0)\) is continuous in \(x\) for every \(y_0 \in Y\) and \(b(x_0, y)\) is continuous in \(y\) for every \(x_0 \in X\), then \(b\) is continuous at the origin \((0,0)\). This is because the product of small perturbations in both \(x\) and \(y\) will result in small perturbations in \(b(x, y)\).

Key concepts

Banach Spaces

Banach spaces are a fundamental concept in functional analysis. They are complete normed vector spaces, meaning that every Cauchy sequence in the space converges to a limit within the space. This property of completeness is vital for various convergence theorems used in analysis.
Here's a breakdown of key features:

• Complete: Every Cauchy sequence converges.
• Normed: There is a function \(orm{.}\) that assigns a non-negative length or size to each vector.
• Vector space: The space supports vector addition and scalar multiplication.

Examples of Banach spaces include the space of continuous functions on a closed interval with the maximum norm, and the space of square-integrable functions with the L2 norm. In the context of the exercise, X and Y are Banach spaces wherein bilinear forms and continuity properties are analyzed.

Bilinear Form

A bilinear form is a function that takes two arguments from two vector spaces and returns a scalar, preserving linearity in each argument when the other is held fixed.
For instance, if \(X\) and \(Y\) are Banach spaces, a bilinear form \(b\) on \(X \oplus Y\) satisfies:

• \(b(x, y_0)\) is linear in \(x\) for every fixed \(y_0 \in Y\).
• \(b(x_0, y)\) is linear in \(y\) for every fixed \(x_0 \in X\).

To represent a bilinear form formally: \(b: X \times Y \rightarrow \mathbb{R}\) where \(b(x, y)\) is linear in each coordinate.
In our exercise, we analyze how different continuity properties relate to each other for such bilinear forms.

Continuity in Functional Analysis

Continuity in functional analysis involves understanding how small changes in the input of a function affect the output. Specifically, a function is continuous if small changes in the input result in small changes in the output.
In the context of bilinear forms on Banach spaces, several types of continuity are important:

• Continuity at the origin: The bilinear form is continuous at the point \((0,0)\).
• Uniform continuity: The bilinear form maintains a consistent rate of change across bounded sets.
• Separate continuity: Continuity in each variable when the other is fixed.

Each type of continuity has implications for how the bilinear form behaves and can lead to other properties or forms of continuity, as shown in the exercise.
Understanding these forms helps in analyzing and working with bilinear forms on Banach spaces effectively.

Uniform Continuity

Uniform continuity is a stronger form of continuity that requires the function to be uniformly continuous across its entire domain.
For a bilinear form \(b\) on Banach spaces \(X\) and \(Y\), uniform continuity on bounded sets means there exists a constant \(K > 0\) such that:
\(|b(x, y)| \leq K \orm{x} \orm{y}\)
for all \(x \in X\) and \(y \in Y\). This ensures that the bilinear form does not grow faster than the product of the norms of its inputs across any bounded set.
This property leads directly to the implication that if a bilinear form is uniformly continuous on bounded sets, it is also globally bounded by a constant multiple of the product of the norms of its arguments, as shown in Step 3 of the solution.

Separate Continuity

Separate continuity for bilinear forms means that holding one variable fixed, the function is continuous in the other variable.
For the bilinear form \(b\), this means:

• \(b(x, y_0)\) is continuous in \(x\) for every fixed \(y_0\).
• \(b(x_0, y)\) is continuous in \(y\) for every fixed \(x_0\).

Separate continuity leads to overall continuity properties at specific points.
In the exercise, we see that separate continuity implies continuity at the origin, as detailed in Step 5. This connection between types of continuity is critical in understanding the behavior of bilinear forms in functional analysis.