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Product Metric Space

Last updated Nov 1, 2022

# Definition

Let $(M_{i}, d_{i}){i \in I}$ be Metric Space

s with Index Set $I$ and let $M = \prod\limits{i \in I}^{} M_{i}$. The Product Metric Space is

M

paired with the Supremum Metric.

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Metrics are Continuous

Metrics are Continuous

...d(x, y).$$ so $d(x,y) - d(x',y') < 2 \delta$. Therefore $|d(x,y) - d(x',y')| < 2 \delta = \epsilon$. Therefore, by the definition of , $\lim\limits_{(x',y') \to (x,y)} d(x',y') = d(x,y)$ . Since $x,y \in M$ were arbitrary, $d$ is a .......

11/7/2022
Product Metric Space

Product Metric Space

Definition Let $(M{i}, d{i}){i \in I}$ be s with $I$ and let $M = \prod\limits{i \in I}^{} M_{i}$ . The is $M$ paired with the ....

11/7/2022
Components Converge iff Convergence in Product Metric Space

Components Converge iff Convergence in Product Metric Space

...\in I}d{i}(x^{(n)}{i}, x{i}) 0 $. There exists$ N \in \mathbb{N} $so for all$ n \geq N $and$ i \in I $,$ d{i}(x^{(n)}{i}, x{i}) < \epsilon $. Thus$ \sup\limits{i \in I}d{i}(x^{(n)}{i}, x_{i}) < \epsilon $for all$ n \geq N $. Therefore$ x^{(n)} \to x $.$ \checkmark$......

11/7/2022

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