Question

Show that if \(\mathbf{X}=\left\\{x_{i}\right\\}_{i=1}^{\infty} \in \ell_{p}\) and \(\mathbf{Y}=\left\\{y_{i}\right\\}_{i=1}^{\infty} \in \ell_{q},\) where \(1 / p+1 / q=1,\) then \(\mathbf{Z}=\left\\{x_{i} y_{i}\right\\} \in \ell_{1}\)

Short answer

Question: Prove that if \(\mathbf{X}=\left\{x_{i}\right\}_{i=1}^{\infty} \in \ell_{p}\) and \(\mathbf{Y}=\left\{y_{i}\right\}_{i=1}^{\infty} \in \ell_{q},\) where \(1 / p+1 / q=1,\) then \(\mathbf{Z}=\left\{x_{i} y_{i}\right\} \in \ell_{1}\). Answer: By applying Hölder's inequality, we can conclude that \(\sum_{i=1}^{\infty}\left|x_iy_i\right| < \infty\), which implies that the vector \(\mathbf{Z}\) with elements \(x_iy_i\) is an element of \(\ell_1\).

Step-by-step solution

Check Conditions for Hölder's Inequality

We want to apply Hölder's inequality, so let's confirm that both vectors are indeed in their respective spaces \(\ell_p\) and \(\ell_q\). By definition, this means that \(\sum_{i=1}^{\infty}|x_i|^p < \infty\) and \(\sum_{i=1}^{\infty}|y_i|^q < \infty\). Since \(\mathbf{X} \in \ell_p\) and \(\mathbf{Y} \in \ell_q\), this condition is true.

Apply Hölder's Inequality

Now we can apply Hölder's inequality to the product of vectors X and Y: $$ \sum_{i=1}^{\infty}\left|x_iy_i\right| \leq \left(\sum_{i=1}^{\infty}|x_i|^p\right)^{1/p}\left(\sum_{i=1}^{\infty}|y_i|^q\right)^{1/q} $$

Use Given Requirements

Since \(1 / p + 1 / q = 1,\) we can substitute \(q = \frac{p}{p - 1}\) in the inequality obtained in Step 2: $$ \sum_{i=1}^{\infty}\left|x_iy_i\right| \leq \left(\sum_{i=1}^{\infty}|x_i|^p\right)^{1/p}\left(\sum_{i=1}^{\infty}|y_i|^{\frac{p}{p-1}}\right)^{\frac{p-1}{p}} $$

Factor out the terms

Notice that we can write the inequality in a slightly different form: $$ \sum_{i=1}^{\infty}\left|x_iy_i\right| \leq \left[\left(\sum_{i=1}^{\infty}|x_i|^p\right)\left(\sum_{i=1}^{\infty}|y_i|^{\frac{p}{p-1}}\right)^{p-1}\right]^{1/p} $$

Recognizing the Convergence Condition

Since \(\mathbf{X} \in \ell_p\) and \(\mathbf{Y} \in \ell_q\), we know that \(\sum_{i=1}^{\infty}|x_i|^p <\infty\) and \(\sum_{i=1}^{\infty}|y_i|^q < \infty\). Because \(1/p + 1/q = 1\) and \(q = p/(p - 1)\), we see that the right-hand side of the inequality is finite.

Proving the Sum Converges

Since the right-hand side of the inequality from Step 5 is finite, we can conclude that the left-hand side is also finite: $$ \sum_{i=1}^{\infty}\left|x_iy_i\right| < \infty $$ This implies that the vector \(\mathbf{Z}=\left\{x_{i} y_{i}\right\} \in \ell_{1}\), which completes the proof.

Key concepts

Convergence

In mathematics, especially in real analysis and functional analysis, convergence is a fundamental concept. It helps us understand the behavior of sequences and series. When we say a series converges, we mean that the sum of its terms approaches a specific number, called the limit, as more terms are added.

For example, consider the sequence of partial sums for a series \(\sum_{i=1}^{\infty}a_i\). If these partial sums get closer and closer to a certain value, then the series is said to converge to that value. Convergence is crucial in determining whether a mathematical expression makes sense or not.

In the context of the exercise, convergence plays a key role. Given the sequences \(\mathbf{X}\) and \(\mathbf{Y}\) in their respective \(\ell_p\) and \(\ell_q\) spaces, their behavior is evaluated through the convergence of their respective power series.

• If the sum \(\sum_{i=1}^{\infty}|x_i|^p < \infty\) and \(\sum_{i=1}^{\infty}|y_i|^q < \infty\), they converge and belong in their respective spaces \(\ell_p\) and \(\ell_q\).


• Step 5 of the solution highlights the importance of convergence, confirming that the series of products \(\sum_{i=1}^{\infty}|x_i y_i|\) also converges. Thus, it belongs in \(\ell_1\).

Lp Spaces

The \(\ell_p\) spaces, also known as \"lp spaces\", are fundamental in the study of functional analysis. They consist of all sequences whose p-th powers are summable. In simple terms, a sequence is in \(\ell_p\) if the sum of its elements raised to the power p is finite.

These spaces have unique properties and structures, characterized by the parameter p, where typically \(1 \leq p < \infty\). The choice of p affects the geometry and other features of these spaces. In the exercise, the \(\ell_p\) space \(\mathbf{X}\) belongs to is key to establishing the convergence of the product series.

• For instance, if \(\mathbf{X} = \{x_i\}\) is in \(\ell_p\), it means that \(\sum_{i=1}^{\infty}|x_i|^p < \infty\).


• Similarly, for \(\mathbf{Y}\) in \(\ell_q\), \(\sum_{i=1}^{\infty}|y_i|^q < \infty\).

The relation between p and q indicated by \(1/p + 1/q = 1\) is critical in the application of Hölder's inequality, which allows proving how the sequence of products ends up in \(\ell_1\). \(\ell_p\) spaces, with their unique properties, are essential for various mathematical analyses.

Real Analysis

Real analysis is the branch of mathematics dealing with real numbers and the real-valued sequences and functions. It provides a rigorous framework for results in calculus and, more generally, mathematical analysis.

A cornerstone of real analysis is understanding convergence, limits, and continuity, all of which are crucial in proving more complex results, like those in the exercise involving Hölder's inequality.

• Hölder's inequality is a powerful tool within real analysis used to show that the sum of products of sequences can be bounded in a useful way.


• This inequality takes advantage of the parameter relationship \(1/p + 1/q = 1\), showing the interplay between \(\ell_p\) and \(\ell_q\) spaces.

Real analysis helps us explore functions' properties, ensuring results remain valid no matter how complex the sequences or functions we are dealing with.

In proving the original exercise, the core principles of real analysis ensure that convergence arguments and applications of established inequalities are rigorous, allowing us to conclude that the combination sequence \(\left\{x_{i} y_{i}\right\}\) indeed belongs to \(\ell_1\). This kind of proof is foundational in real analysis, demonstrating how properties of spaces and inequalities join together to reach a valid conclusion.