Question

Suppose that \(\left(A_{i}, \rho_{i}\right), 1 \leq i \leq k,\) are metric spaces. Let $$ A=A_{1} \times A_{2} \times \cdots \times A_{k}=\left\\{\mathbf{X}=\left(x_{1}, x_{2}, \ldots, x_{k}\right) \mid x_{i} \in A_{i}, 1 \leq i \leq k\right\\} $$ If \(\mathbf{X}\) and \(\mathbf{Y}\) are in \(A,\) let $$ \rho(\mathbf{X}, \mathbf{Y})=\sum_{i=1}^{k} \rho\left(x_{i}, y_{i}\right) $$ (a) Show that \(\rho\) is a metric on \(A\). (b) Let \(\left\\{\mathbf{X}_{r}\right\\}_{r=1}^{\infty}=\left\\{\left(x_{1 r}, x_{2 r}, \ldots, x_{k r}\right)\right\\}_{r=1}^{\infty}\) be a sequence in \(A .\) Show that $$ \lim _{r \rightarrow \infty} \mathbf{X}_{r}=\widehat{\mathbf{X}}=\left(\widehat{x}_{1}, \widehat{x}_{2}, \ldots, \widehat{x}_{k}\right) $$ if and only if $$ \lim _{r \rightarrow \infty} x_{i r}=\widehat{x}_{i}, \quad 1 \leq i \leq k $$ (c) Show that \(\left\\{\mathbf{X}_{r}\right\\}_{r=1}^{\infty}\) is a Cauchy sequence in \((A, \rho)\) if and only if \(\left\\{x_{i} r\right\\}_{r=1}^{\infty}\) is a Cauchy sequence in \(\left(A_{i}, \rho_{i}\right), 1 \leq i \leq k\). (d) Show that \((A, \rho)\) is complete if and only if \(\left(A_{i}, \rho_{i}\right)\) is complete, \(1 \leq i \leq k\).

Short answer

In summary, we analyzed a metric space formed by the product of \(k\) metric spaces \((A_i, \rho_i)\). We first proved that a defined distance function \(\rho\) on this composite space is indeed a metric. Then, we connected the convergence and Cauchy sequences in this new metric space with the convergence and Cauchy sequences in each component metric space \((A_i, \rho_i)\). Lastly, we established the relationship between the completeness of the composite space \((A, \rho)\) and the completeness of each component metric space \((A_i, \rho_i)\).

Step-by-step solution

a) Show that \(\rho\) is a metric on \(A\)

In order to prove that \(\rho\) is a metric on \(A\), we need to show that it fulfills the properties of a metric: 1. Non-negativity: \(\rho(\mathbf{X}, \mathbf{Y}) \geq 0\) 2. Identity: \(\rho(\mathbf{X}, \mathbf{Y}) = 0\) if and only if \(\mathbf{X} = \mathbf{Y}\) 3. Symmetry: \(\rho(\mathbf{X}, \mathbf{Y}) = \rho(\mathbf{Y}, \mathbf{X})\) 4. Triangle inequality: \(\rho(\mathbf{X}, \mathbf{Z}) \leq \rho(\mathbf{X}, \mathbf{Y}) + \rho(\mathbf{Y}, \mathbf{Z})\) Since \(\rho_i\) are metrics on each \(A_i\), we can apply the properties of these individual metrics to show that \(\rho\) is a metric on \(A\).

b) Convergence in \(A\) in terms of convergence in \(A_i\)

The task is to prove that \(\lim_{r \rightarrow \infty}\mathbf{X}_{r} = \widehat{\mathbf{X}}\) if and only if \(\lim_{r \rightarrow \infty} x_{ir} = \widehat{x}_i\), for all \(1 \leq i \leq k\). We are given that \(\mathbf{X}_r \in A\), and that \(\widehat{\mathbf{X}} \in A\). We proceed by splitting the proof into two parts: 1. If \(\lim_{r \rightarrow \infty}\mathbf{X}_{r} = \widehat{\mathbf{X}}\), then \(\lim_{r \rightarrow \infty} x_{ir} = \widehat{x}_i\) for \(1 \leq i \leq k\). 2. If \(\lim_{r \rightarrow \infty} x_{ir} = \widehat{x}_i\) for \(1 \leq i \leq k\), then \(\lim_{r \rightarrow \infty}\mathbf{X}_{r} = \widehat{\mathbf{X}}\).

c) Cauchy sequences in \((A, \rho)\) and \((A_i, \rho_i)\)

The task is to prove that \(\{\mathbf{X}_r\}_{r=1}^{\infty}\) is a Cauchy sequence in \((A, \rho)\) if and only if \(\{x_{ir}\}_{r=1}^{\infty}\) is a Cauchy sequence in \((A_i, \rho_i)\), for all \(1 \leq i \leq k\). To show this, we first recall the definition of a Cauchy sequence. A sequence \(\{x_n\}\) in a metric space \((X,d)\) is called a Cauchy sequence if, given any \(\varepsilon > 0\), there exists a positive integer \(N\) such that for all \(m, n \geq N\), we have \(d(x_m, x_n) < \varepsilon\). We split the proof into two parts: 1. If \(\{\mathbf{X}_r\}_{r=1}^{\infty}\) is a Cauchy sequence in \((A, \rho)\), then \(\{x_{ir}\}_{r=1}^{\infty}\) is a Cauchy sequence in \((A_i, \rho_i)\) for all \(1 \leq i \leq k\). 2. If \(\{x_{ir}\}_{r=1}^{\infty}\) is a Cauchy sequence in \((A_i, \rho_i)\) for all \(1 \leq i \leq k\), then \(\{\mathbf{X}_r\}_{r=1}^{\infty}\) is a Cauchy sequence in \((A, \rho)\).

d) Completeness of \((A, \rho)\) and \((A_i, \rho_i)\)

Finally, we need to show that \((A, \rho)\) is complete if and only if \((A_i, \rho_i)\) is complete for all \(1 \leq i \leq k\). Recall that a metric space \((X, d)\) is said to be complete if every Cauchy sequence in \(X\) has a limit in \(X\). We now split the proof into two parts: 1. If \((A, \rho)\) is complete, then \((A_i, \rho_i)\) is complete for all \(1 \leq i \leq k\). 2. If \((A_i, \rho_i)\) is complete for all \(1 \leq i \leq k\), then \((A, \rho)\) is complete.

Key concepts

Real Analysis

Real analysis is a branch of mathematics that deals with real numbers and the functions defined on them. It encompasses the rigorous study of sequences, series, and continuity, along with the behavior of functions and the convergence of sequences.

The concept of a metric space fits within real analysis, providing a framework for discussing distance and convergence not only on the real numbers but on more complex structures. In real analysis, a metric space is any set where we can talk about the distance between elements. This abstract concept extends to many other areas such as functional analysis and topology, building a foundation for understanding continuity, limits, and the notion of closeness in a broader sense.

Real analysis emphasizes proofs and logical reasoning. For any property or theorem, such as the definition of a convergent sequence or a continuous function, real analysis requires a precise demonstration that the statement holds under well-defined conditions. These proofs serve as the backbone of mathematical rigor, ensuring that the calculations and applications built on these concepts are sound.

Cauchy Sequences

Cauchy sequences are a fundamental concept in real analysis, named after the French mathematician Augustin-Louis Cauchy. They are sequences where the elements become arbitrarily close to each other as the sequence progresses.

A sequence \(x_n\) of points in a metric space \(X, d\) is a Cauchy sequence if for every positive number \(\epsilon > 0\), there exists an integer \(N\) such that for all natural numbers \(m, n > N\), the distance \(d(x_m, x_n) < \epsilon\). This definition focuses on the idea that the 'tail end' of the sequence doesn't spread out but stays bunched together, even though the sequence might not necessarily converge to a specific point within the space.

Importantly, the existence of a limit for a Cauchy sequence is guaranteed only if the metric space is complete. Thus, identifying a sequence as Cauchy is a step towards understanding its convergence behavior, contingent on the completeness of the underlying metric space.

Metric Space Completeness

What Makes a Metric Space Complete?

A metric space is called complete if every Cauchy sequence in the space converges to a limit that is within the space. In essence, it has no 'holes'; sequences that get increasingly closer to some point actually have that point within the same space.

In the context of the given exercise, a product space \(A\) formed by the Cartesian product of individual metric spaces \(A_i\), with the metric \(\rho\) defined as the sum of the individual distances, inherits the completeness property from the factors spaces \(A_i\). Therefore, if each \(A_i\) is complete, the product space \(A\) is also complete.

This property is crucial, for instance, in functional analysis where we deal with spaces of functions and need to know if every Cauchy sequence of functions has a limit function in the same space. It underpins much of the work in analysis and related fields because it ensures that 'limits of limits' make sense in the space.

Convergence in Metric Spaces

Convergence in metric spaces extends the familiar concept of a sequence getting closer to a point from real numbers to the context of an abstract set with a notion of distance. A sequence \(\{x_n\}\) in a metric space \(X, d\) is said to converge to a point \(x \in X\) if, for every \(\epsilon > 0\), there is an integer \(N\) such that for all \(n > N\), \(d(x_n, x) < \epsilon\).

In the exercise provided, the convergence of sequences in the product space \(A\) is characterized by the convergence of their respective components in each of the metric spaces \(A_i\). This implies that analyzing convergence in a complex structure can be broken down into evaluating convergence in simpler, more familiar spaces.

Understanding convergence in metric spaces allows mathematicians to generalize the concept of a limit, which is pivotal for defining continuous functions and integral calculus in more abstract settings than just the real numbers.