Metric space
High school
A metric space is a set of points with a way of measuring the distance between every two points.
Undergraduate
A metric space is a set $X$ together with a metric $d$ on $X$ that defines the distance between any two points.
Examples
Examples
- $\reals^n$ with the Euclidean norm Undergraduate
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Continuous functions on $[0, 1]$ with the max norm
Graduate
The continuous functions on $[0, 1]$ with the max norm form a metric space with $d(f, g) = \norm{f - g}$ because every normed space is a metric space.
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Polynomials on $[0, 1]$ with the max norm
Graduate
The polynomials on $[0, 1]$ with the max norm form a metric space, because every normed space is a metric space, with $d(p, q) = \norm{p - q}$.
Notes
A metric space only requires a notion of distance, but does not require addition, or scaling, lengths of vectors, or angles. In other words, a metric space does not need to be a vector space (scaling+addition), a normed vector space (lengths), or an inner product space (angles).