Metric on sequences and functions

Metric on Sequences and Functions A metric on a sequence (or function) is a function that measures the distance between points in the sequence or function s...

Metric on Sequences and Functions

A metric on a sequence (or function) is a function that measures the distance between points in the sequence or function space. This allows us to compare the lengths of sequences and the heights of functions.

Formal Definition:

Let (d) be a metric on a sequence (X). A sequence (x_n) is said to be (\varepsilon-metrically\ similar) to a sequence (y_n) if the distance between them, measured by (d), is less than (\varepsilon). That is, if

$\lim_{n\to\infty}d(x_n, y_n) = 0$

for all (\varepsilon>0).

Examples of Metric Spaces:

Euclidean metric: For sequences of real numbers, the Euclidean metric is defined by the distance between points to be the square root of the sum of the squared differences between their coordinates.

$d(x_n, y_n) = \sqrt{(x_n - y_n)^2}$

Maximum metric: For sequences of real numbers, the maximum metric is defined by the distance between points to be the maximum difference between their coordinates.

$d(x_n, y_n) = \max\{|x_n - y_n| | n\in\mathbb{N}}$

Uniform metric: For functions, the uniform metric is defined by the distance between points to be the maximum difference between their values.

$d(f_n, f_m) = \max\{|f_n - f_m| | n,m\in\mathbb{N}}$

Key Concepts:

A metric defines a metric space, which is a set of points with a distance function.
A sequence is said to be convergent if its limit is a point in the metric space.
A function is said to be continuous if its limit is equal to its value at each point in the domain