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Normed vector spaces

A normed vector space is a vector space equipped with a norm, which is a function that measures the length of vectors. The norm of a vector v is typically denoted by ‖ v ‖ {\displaystyle \lVert v\rVert }. Any normed vector space can be equipped with a metric in which the distance between two vectors x and y is given by d (x, y):= ‖ x − y ‖. {\displaystyle d(x,y):=\lVert x-y\rVert.} The metric d is said to be induced by the norm ‖ ⋅ ‖ {\displaystyle \lVert {\cdot }\rVert }. Conversely, if a metric d on a vector space X is

then it is the metric induced by the norm ‖ x ‖:= d (x, 0). {\displaystyle \lVert x\rVert :=d(x,0).} A similar relationship holds between seminorms and pseudometrics.

Among examples of metrics induced by a norm are the metrics d, d, and d on R 2 {\displaystyle \mathbb {R} ^{2}}, which are induced by the Manhattan norm, the Euclidean norm, and the maximum norm, respectively. More generally, the Kuratowski embedding allows one to see any metric space as a subspace of a normed vector space.

Infinite-dimensional normed vector spaces, particularly spaces of functions, are studied in functional analysis. Completeness is particularly important in this context: a complete normed vector space is known as a Banach space. An unusual property of normed vector spaces is that linear transformations between them are continuous if and only if they are Lipschitz. Such transformations are known as bounded operators.