Normed Space

We ll generalize from euclidean spaces to more general spaces such as spaces of functions.

Normed space. Theorem 3 7 examples of banach spaces 1every finite dimensional vector space x is a banach space. A normed space is a pair x consisting of a vector space x over a scalar field k together with a distinguished norm. De nition 2 a vector space v is a normed vector space if there is a norm function mapping v to the non negative real numbers written kvk. Kvk 0 8v 2 v and kvk 0 if and only if v 0.

Norms and metrics normed vector spaces and metric spaces we re going to develop generalizations of the ideas of length or magnitude and distance. K vk j jkvk. D x y 0 if and only if x y d x y d y x d x y d x z d z y. For v 2 v.

And satisfying the following 3 axioms. Defn a metric space is a pair x d where x is a set and d. 8v 2 v and8 2 r. Norm normed ring vector space cite this as.

X 2 0 with the properties that for each x y z in x. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of length in the real world. Definition banach space a banach space is a normed vector space which is also complete with respect to the metric induced by its norm. A vector space x is said to be finite dimensional if there is a positive integer.

Like all norms this norm induces a translation invariant distance function called the canonical or induced metric defined by d x y y x for all vectors x y x. 2the sequence space ℓpis a banach space for any 1 p. Normed space from mathworld a wolfram web resource. A vector space possessing a norm.

Here j j absolute value of. An arbitrary subset m of a vector space x is said to be linearly independent if every non empty finite subset of m is linearly independent.