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A vector space with a specified norm is called a normed vector space. In a similar manner, a vector space with a seminorm is called a seminormed vector space. The term pseudonorm has been used for several related meanings. It may be a synonym of "seminorm". [1]
A norm induces a distance, called its (norm) induced metric, by the formula (,) = ‖ ‖. which makes any normed vector space into a metric space and a topological vector space. If this metric space is complete then the normed space is a Banach space .
Suppose a vector norm ‖ ‖ on and a vector norm ‖ ‖ on are given. Any matrix A induces a linear operator from to with respect to the standard basis, and one defines the corresponding induced norm or operator norm or subordinate norm on the space of all matrices as follows: ‖ ‖, = {‖ ‖: ‖ ‖ =} = {‖ ‖ ‖ ‖:} . where denotes the supremum.
This definition can be formalized in Cartesian space by defining the dot product and specifying that two vectors in the plane are orthogonal if their dot product is zero. Similarly, the construction of the norm of a vector is motivated by a desire to extend the intuitive notion of the length of a vector to higher-dimensional spaces.
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in ^ (pronounced "v-hat"). The normalized vector û of a non-zero vector u is the unit vector in the direction of u, i.e.,
By definition, all Euclidean vectors have a magnitude (see above). However, a vector in an abstract vector space does not possess a magnitude. A vector space endowed with a norm, such as the Euclidean space, is called a normed vector space. [8] The norm of a vector v in a normed vector space can be considered to be the magnitude of v.
Uniform norms are defined, in general, for bounded functions valued in a normed space.Let be a set and let (, ‖ ‖) be a normed space.On the set of functions from to , there is an extended norm defined by
The norm of a vector is represented with double bars on both sides of the vector. The norm of a vector v can be represented as: ‖ ‖ The norm is also sometimes represented with single bars, like | |, but this can be confused with absolute value (which is a type of norm).