| Revision as of 20:32, 13 August 2001 editLoisel (talk | contribs)Extended confirmed users1,558 edits dual space of a normed space | Revision as of 13:04, 16 August 2001 edit undoZundark (talk | contribs)Extended confirmed users, File movers, Pending changes reviewers29,653 edits import some text from Linear_Algebra/Normed_Vector_SpaceNext edit → | ||
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| A '''normed vector space''' (or simply '''normed space''') is a ] ''V'' over a ] ''K'' (which must be either the ] or the ]) together with a ] (called a '''norm''') that associates to each ''x'' in ''V'' a real number denoted by ||''x''||. The norm must have the following properties, for all ''a'' in ''K'' and all ''x'' and ''y'' in ''V''. | |||
| Let V be a vector space. | |||
| # ||''x''|| >= 0, with equality if and only if ''x'' = 0. | |||
| # ||''ax''|| = |''a''|.||''x''||. | |||
| # ||''x''+''y''|| <= ||''x''|| + ||''y''||. (''The triangle inequality.'') | |||
| A familiar example is the space '''R'''<sup>''n''</sup> (where '''R''' denotes the real numbers and ''n'' is any ]) with ||''x''|| being the Euclidean distance of ''x'' from the origin. | |||
| For any normed space we can define the distance between two vectors as ||''x''-''y''||. | |||
| A function f:V→R (R being the set of real numbers) is called a norm when it satisfies the following axioms: | |||
| This makes the normed space into a ]. | |||
| If this metric space is complete then the normed space is called a ]. | |||
| ⚫ | ], a ] of normed vector spaces would be a linear map that preserves the norm. This isn't very useful, so a notion which may be more appropriate to ] is often used: a homomorphism is a linear map that is continuous. When referring to a norm-preserving linear map, the term isometry is used. Note that an isometry is automatically an ] (its inverse is an isometry as well.) When speaking of isomorphisms of normed spaces, one normally means an isometry, or at the very least a continuous, onto linear map with a continuous inverse. | ||
| ⚫ | When speaking of normed vector spaces, we augment the notion of dual (see ]) to also include the norm. The dual V<sup>*</sup> of a normed vector space V is the space of all continuous linear maps from V to the root field (the complexes or the reals) -- such linear maps are labeled "functionals". This continuity requirement destroys the self-duality property that ordinary vector spaces enjoy. Note that the norm of a functional F is defined by the sup of |F(x)| where x ranges over unit vectors in V. | ||
| #If α is a non-negative real number then f(αv)=αf(v) for any v in V. | |||
| #For any vectors u,v in V, f(u+v)≤f(u)+f(v) | |||
| #f(v)\ge;0 for any vector v in V | |||
| #f(v)=0 if and only if v=0 | |||
| The norm of a vector v is usually denoted by ||v||. | |||
| A norm is useful for measuring distance. The distance between u and v is defined to be ||u-v|| | |||
| An open ball is a set of the form { u in V; ||u-v||<ε } for some fixed v in V and ε>0 in R. The set of balls defines a ] for a ]. In other words, if v<sub>k</sub> is a sequence of vectors then we say that v<sub>k</sub> converges to v if and only if the real number ] ||v<sub>k</sub>-v|| converges to zero. | |||
| ⚫ | Categorically speaking, a homomorphism of normed vector spaces would be a linear map that preserves the norm. This isn't very useful, so a notion which may be more appropriate to ] is often used: a homomorphism is a linear map that is continuous. When referring to a norm-preserving linear map, the term isometry is used. Note that an isometry is automatically an isomorphism (its inverse is an isometry as well.) When speaking of isomorphisms of normed spaces, one normally means an isometry, or at the very least a continuous, onto linear map with a continuous inverse. | ||
| ⚫ | When speaking of normed vector spaces, we augment the notion of dual (see ]) to also include the norm. The dual V<sup>*</sup> of a normed vector space V is the space of all continuous linear maps from V to the root field (the complexes or the reals) -- such linear maps are labeled "functionals |
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Revision as of 13:04, 16 August 2001
A normed vector space (or simply normed space) is a vector space V over a field K (which must be either the real numbers or the complex numbers) together with a function (called a norm) that associates to each x in V a real number denoted by ||x||. The norm must have the following properties, for all a in K and all x and y in V.
- ||x|| >= 0, with equality if and only if x = 0.
- ||ax|| = |a|.||x||.
- ||x+y|| <= ||x|| + ||y||. (The triangle inequality.)
A familiar example is the space R (where R denotes the real numbers and n is any natural number) with ||x|| being the Euclidean distance of x from the origin.
For any normed space we can define the distance between two vectors as ||x-y||. This makes the normed space into a metric space. If this metric space is complete then the normed space is called a Banach space.
Categorically speaking, a homomorphism of normed vector spaces would be a linear map that preserves the norm. This isn't very useful, so a notion which may be more appropriate to topological vector spaces is often used: a homomorphism is a linear map that is continuous. When referring to a norm-preserving linear map, the term isometry is used. Note that an isometry is automatically an isomorphism (its inverse is an isometry as well.) When speaking of isomorphisms of normed spaces, one normally means an isometry, or at the very least a continuous, onto linear map with a continuous inverse.
When speaking of normed vector spaces, we augment the notion of dual (see dual space) to also include the norm. The dual V of a normed vector space V is the space of all continuous linear maps from V to the root field (the complexes or the reals) -- such linear maps are labeled "functionals". This continuity requirement destroys the self-duality property that ordinary vector spaces enjoy. Note that the norm of a functional F is defined by the sup of |F(x)| where x ranges over unit vectors in V.