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 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
for all (\varepsilon>0).
Examples of Metric Spaces:
d(x_n, y_n) = \max\{|x_n - y_n| | n\in\mathbb{N}}
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