Metric Spaces Lecture Notes

Vocabulary and definitions regarding the fundamental concepts and specific examples of metric spaces as presented in the Math 127C lecture.

Metric

A function $d$ on any set $A$ such that $d : A \times A \rightarrow \text{R}$ and the following four properties hold for all $x, y, z \text{ in } A$: $M1$ (Self-Distance), $M2$ (Positivity), $M3$ (Symmetry), and $M4$ (Triangle Inequality).

Metric Space

An ordered pair $(A, d)$ where $A$ is a set and $d$ is a metric.

M1 (Self-Distance)

The property of a metric where the distance from a point to itself is zero, defined as $d(x, x) = 0$.

M2 (Positivity)

The property of a metric where if $x \neq y$, then $d(x, y) > 0$.

M3 (Symmetry)

The property of a metric where the distance between two points is independent of order, defined as $d(x, y) = d(y, x)$.

M4 (Triangle Inequality)

The property of a metric where the distance between two points is less than or equal to the sum of the distances via a third point, defined as $d(x, y) \neq d(x, z) + d(z, y)$.

Discrete Metric

A valid metric paired with any set $A$ where the function is defined as $d(x, y) = 0$ if $x = y$ and $d(x, y) = 1$ if $x \neq y$.

Standard Metric on $R$

A metric space where $A = R$ and the distance function is defined by the absolute value of the difference: $d(x, y) = |x - y|$.

Taxicab Metric on $R^n$

A metric on $A = R^n$ defined by $d(x, y) = \text{sum}_{i=1}^n (|x_i - y_i|)$, representing a rectilinear path where only $90$ degree turns are allowed.

Euclidean Metric on $R^n$

The "usual" distance function on $A = R^n$ defined by $d(x, y) = (\text{sum}_{i=1}^n (|x_i - y_i|^2))^{\frac{1}{2}}$.

Chebyshev Metric on $R^n$

A metric on $A = R^n$ defined by $d(x, y) = \text{sup}_{i=1}^n (|x_i - y_i|)$, which identifies the distance between two points as the largest single coordinate difference.

Supremum vs. Maximum in Chebyshev Metric

In the definition of the Chebyshev Metric where $n$ is finite, the supremum is always equal to the maximum.