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# 数学代写|数学分析作业代写Mathematical Analysis代考|MTH131LR Product Spaces

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## 数学代写|数学分析作业代写MATHEMATICAL ANALYSIS代考|Product Spaces

The Euclidean plane $\mathbb{R}^{2}$, as the product of two copies of $\mathbb{R}$, is the simplest example of a product space. We saw in section $4.3$ that the Euclidean metric in the plane, although the most natural, is equivalent to several other metrics, including the $\infty$-metric, which, according to the definition below, is the product metric on $\mathbb{R}^{2}$. It is only natural to expect that the product of two open intervals should be an open subset of $\mathbb{R}^{2}$, and the definition we adopt for the product metric smoothly guarantees that. When we identify the complex field with $\mathbb{R}^{2}$, the convergence of a complex sequence $z_{n}=x_{n}+i y_{n}$ is equivalent to the convergence of its real and imaginary parts in $\mathbb{R}$, and one expects that product metrics in general should extend this property. Not only does the product metric preserve the componentwise convergence in the factor spaces, it is characterized by it. You will see that the product metric is the weakest metric that guarantees componentwise convergence in the factor spaces. Additionally, we will show that the product metric admits the continuity of the projections on the factor spaces and, once again, is characterized by it. We therefore think of the product metric as the most economical metric that generalizes the properties of Euclidean space in relation to its factor spaces.

Let $\left{\left(X_{i}, d_{i}\right)\right}_{i=1}^{n}$ be a finite set of metric spaces, and let $X=\prod_{i=1}^{n} X_{i}=$ $\left{\left(x_{1}, \ldots, x_{n}\right): x_{i} \in X_{i}\right}$ be the Cartesian product of the underlying sets $X_{i}$.
Definition. The product metric $D$ on $X$ is defined by
$$D(x, y)=\max {1 \leq i \leq n} d{i}\left(x_{i}, y_{i}\right) .$$
Here $x=\left(x_{1}, \ldots, x_{n}\right)$ and $\left(y_{1}, \ldots, y_{n}\right)$ are points in $X$. The verification that $D$ is a metric is straightforward.

## 数学代写|数学分析作业代写MATHEMATICAL ANALYSIS代考|Separable Spaces

Although the rigorous definition of the real line was a giant leap in the development of mathematics, it would not be nearly as useful an invention had it not been for the fact that it contains the rational numbers as a dense subset. Indeed, all practical computations, including machine calculations, are done exclusively using rational numbers. The simplicity of rational numbers is enhanced by their countability. Thus $\mathbb{Q}$ is numerous enough, simple enough, but not too enormous to be a useful approximation of $\mathbb{R}$. It is a reasonable quest to study metric spaces that contain a countable dense subset (of simpler elements). Such spaces are, by definition, separable. You will see that many (but not all) metric spaces are separable. The classical example is the space $\mathcal{C}[0,1]$. It is well known that (see section 4.8) the set of polynomials with rational coefficients, which is countable, is dense in $\mathcal{C}[0,1]$. What can be a nicer approximation of a continuous function than a rational polynomial! Separability of a metric space turns out to be equivalent to the existence of a countable collection of open sets that generate all open sets, which is an added benefit and an important characterization of separability.
Definition. A subset $A$ of a metric space $X$ is dense in $X$ if $\bar{A}=X$. By theorem 4.2.5, $A$ is dense in $X$ if and only if every point in $X$ is the limit of a sequence in A. Equivalently, $A$ is dense in $X$ if and only if for every $x \in X$ and every $\epsilon>0$, there is an element $a \in A$ such that $d(x, a)<\epsilon$.

Example 1. Given a function $f \in \mathcal{C}[0,1]$ and a number $\epsilon>0$, there exists a continuous, piecewise linear function $g$ such that $|f-g|_{\infty}<\epsilon$.

We use the uniform continuity of $f$ (see example 8 on section 1.2). Let $\delta>0$ be such that $|f(x)-f(y)|<\epsilon$ whenever $|x-y|<\delta$. Choose a natural number $n$ such that $1 / n<\delta$, and, for $0 \leq j \leq n$, let $x_{j}=j / n$. Define the function $g$ to be the continuous, piecewise linear function such that $g\left(x_{j}\right)=f\left(x_{j}\right)$ for $0 \leq j \leq n$. By construction, $|f-g|_{\infty}<\epsilon$. Observe that this example says that the space of continuous, piecewise linear functions is dense in $\mathcal{C}[0,1]$.

Definition. A metric space is separable if it contains a countable dense subset.

## 数学代写|数学分析作业代写 MATHEMATICAL ANALYSIS代 考|PRODUCT SPACES

$D(x, y)=\max 1 \leq i \leq n d i\left(x_{i}, y_{i}\right)$

## Matlab代写

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