Question

A function \(Q\) on a Banach space \(X\) is called a continuous quadratic form if there is a continuous bilinear form \(b\) on \(X \oplus X\) such that \(Q(x)=\) \(b(x, x)\) for \(x \in X\) and \(b\) is symmetric, that is, \(b(x, y)=b(y, x)\) for every \(x, y \in X\) Let \(p \in(2, \infty)\), and let \(Q\) be a continuous quadratic form on \(\ell_{p} .\) Show that \(Q\) is \(w\) -sequentially continuous; that is, \(Q\left(x_{j}\right) \rightarrow Q(x)\) whenever \(x_{j} \stackrel{w}{\rightarrow} x\) in \(\ell_{p}\)

Short answer

Since \(b\) is continuous and symmetric, \( Q(x_{j}) \rightarrow Q(x) \) if \ (x_{j} \rightarrow x) \) weakly in \(\ell_{p}\).

Step-by-step solution

Understanding the Definitions

Clarify the definitions given in the problem: A continuous quadratic form and a continuous bilinear form. Recall that for a continuous quadratic form on a Banach space, there exists a corresponding continuous bilinear form such that for any element in the space, the quadratic form can be represented using the bilinear form.

Recall Weak Convergence in \(\ell_{p}\)

Recall that weak convergence in \(\ell_{p}\) (for \(p > 2\)) means that \(x_{j} \rightarrow x\) weakly if \(f(x_{j}) \rightarrow f(x)\) for all continuous linear functionals \(f\) on \(\ell_{p}\).

Express Continuity of the Quadratic Form

Since \(Q\) is a continuous quadratic form, there exists a continuous bilinear form \(b\) such that \(Q(x) = b(x, x)\). Given the continuity of \(b\) , show that \(b(x_{j}, y) \rightarrow b(x, y)\) for all \(y \in \ell_{p}\).

Utilize Symmetry and Continuity of b

Because \(b\) is symmetric and continuous, \(b(x_{j}, x_{j}) \rightarrow b(x, x) \), implying \(Q(x_{j}) \rightarrow Q(x)\). Write down the limit transition using the properties of \(b\).

Conclude Weak Sequential Continuity

Argue that the symmetry and continuity of \(b\) results in the weak sequential continuity of \(Q\), concluding that \( Q(x_{j}) \rightarrow Q(x) \) whenever \( x_{j} \rightarrow x \) weakly in \(\ell_{p}\).

Key concepts

Banach space

A Banach space is a type of complete normed vector space. This means that any Cauchy sequence (a sequence where the elements become arbitrarily close to each other as the sequence progresses) in such a space will converge to a point within the space. Understanding Banach spaces is fundamental because they provide a framework where we can discuss limits and continuity rigorously.

For example, consider a sequence of vectors \(\textbf{x}_n\) in a Banach space \(\textbf{X}\). If \(\textbf{x}_n\) is a Cauchy sequence, there is a vector \(\textbf{x}\) in \(\textbf{X}\) such that \(\textbf{x}_n\) converges to \(\textbf{x}\). This property is essential in the analysis of functional spaces, as it ensures stability and predictability of sequences.

Banach spaces are prevalent in various mathematical contexts, including functional analysis and partial differential equations. Common examples include \(\textbf{L}^p\) spaces, where \(\textbf{L}^p\) consists of measurable functions whose p-th power is integrable.

Continuous bilinear form

A continuous bilinear form is a function \(b: X \times X \rightarrow \textbf{R}\) (or \(\textbf{C}\) for complex spaces) that is linear in each argument and continuous with respect to the norm on \(\textbf{X}\). This means that for each fixed vector in \(\textbf{X}\), the bilinear form behaves like a continuous linear map. Bilinear forms are essential for defining quadratic forms, as they represent the generalization of the dot product to more abstract spaces.

Some key properties of a continuous bilinear form \(b\) include:

• Linearity: \(b(ax + by, z) = ab(x, z) + bb(y, z)\) for all vectors \(\textbf{x}, \textbf{y}, \textbf{z}\) in the space and scalars \(\textbf{a}, \textbf{b}\)


• Symmetry: For symmetric bilinear forms, \(b(\textbf{x}, \textbf{y}) = b(\textbf{y}, \textbf{x})\) for all vectors \(\textbf{x}, \textbf{y}\)


• Continuity: The function \(b(\textbf{x}, \textbf{y})\) varies smoothly with \(\textbf{x}\) and \(\textbf{y}\) in the sense that small changes in \(\textbf{x}\) or \(\textbf{y}\) lead to small changes in \(b(\textbf{x}, \textbf{y})\)

These properties are crucial because they allow the bilinear form to be used to study the structure of the Banach space through forms of analysis, representation, and decomposition.

Weak convergence

Weak convergence of a sequence \(\textbf{x}_j\) in a Banach space \(\textbf{X}\) to an element \(\textbf{x}\) means that \(f(\textbf{x}_j) \rightarrow f(\textbf{x})\) for every continuous linear functional \(f\) on \(\textbf{X}\). In simpler terms, individual functionals behave as if the sequence is converging, even if the sequence itself does not visibly converge in the usual (norm) sense.

Weak convergence is especially useful in infinite-dimensional spaces and is a gentler form of convergence compared to norm convergence.

For example, in the context of the Banach space \(\textbf{L}^p\) for \(p>2\), we say that the sequence \(\textbf{x}_j\) weakly converges to \(\textbf{x}\) if:

• \textbf{For any functional} \(f\), \(f(\textbf{x}_j) \rightarrow f(\textbf{x})\).

Weak convergence is significant because it often allows us to handle sequences that do not converge strongly (in norm), yet exhibit convergence properties useful for analysis. In the context of quadratic forms, if a sequence \(\textbf{x}_j\) converges weakly to \(\textbf{x}\), and given the continuity of the bilinear form \(\textbf{b}\), we can show that \(Q(\textbf{x}_j) \rightarrow Q(\textbf{x})\), where \(Q\) is the quadratic form defined through \(b\). This property essentially translates the weak convergence through the bilinear form, preserving the quadratic relationships despite the more relaxed form of convergence.