Normed vector space

A vector space equipped with a norm, a function assigning each vector a nonnegative size. The norm induces a metric by measuring the norm of the difference between two vectors.

Vector space: A set with vector addition and scalar multiplication satisfying axioms such as associativity and distributivity. The normed structure adds a size function to this algebraic foundation.
Triangle inequality: The inequality stating that the distance between two points cannot exceed the sum of distances through a third. For a norm, it bounds the size of a sum by the sizes of its summands.
Banach space: A complete normed vector space in which every Cauchy sequence converges. Completeness is an additional condition not guaranteed by being normed.
Lp space: A function space whose elements have finite p-th-power integral, or a corresponding sequence-space sum. Its norms measure the size of functions or sequences across a range of applications.
Euclidean norm: The norm of a coordinate vector given by the square root of the sum of its squared coordinates. It is the familiar norm on finite-dimensional real coordinate spaces.
Norm: A function assigning nonnegative lengths to vectors, vanishing only at zero and obeying homogeneity and the triangle inequality. Its axioms define the extra structure that distinguishes this space.
Induced metric: The distance function d(x,y) = ‖x − y‖ obtained from a norm on a vector space. It turns norm measurements into distances between points.
Inner product space: A vector space equipped with an inner product that defines lengths and angles. An inner product gives angle information that a general norm need not provide.
Space of continuous functions: A vector space of continuous functions on a specified domain, often equipped with the supremum norm. The supremum norm turns uniform approximation into distance in a normed space.
Manhattan norm: The coordinate norm that sums the absolute values of a vector’s components. It gives a distinct geometry on the same coordinate space as the Euclidean norm.

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