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| Note that there need not be only one function that is a valid norm. Indeed, generalizing the concept of length allows different norms to be defined on '''R'''<sup> |
Note that there need not be only one function that is a valid norm. Indeed, generalizing the concept of length allows different norms to be defined on '''R'''<sup>''n''</sup>. | ||
Revision as of 01:09, 24 January 2002
With 2 or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive. This can be extended to any vector space R. For more abstract vector spaces, a norm is a generalization of this idea. A vector space on which a norm is defined is then called a normed vector space.
For any vector space V over a field K (which must be either the real numbers or the complex numbers), a norm is a function from V to R, the real numbers — that is, it associates to each vector v in V a real number, which is usually denoted ||v||.
Furthermore, a norm must satisfy the following conditions:
- For all a in K and all u and v in V,
- ||v|| >= 0, with equality if and only if v = 0.
- ||av|| = |a|.||v||.
- ||u+v|| <= ||u|| + ||v||
These conditions essentially demand that the norm behave in the same way that we intuitively expect for it to be a notion of length:
- a vector always has a strictly positive length. The only exception is the zero vector which has length zero.
- multiplying a vector by a number has the same effect on the length
- the Triangle Inequality, which amounts roughly to saying that the distance from A to B to C is never shorter than going directly from A to C.
The intuitive formula for the length of a vector in R or R, as well as its generalisation to R, can be shown to satisfy these conditions. This is often called the Euclidean norm, and sometimes to emphasise that the space is being considered together with this norm, it is written E instead of R. (a normal 'E', not double-barrelled like the 'R'.)
Note that there need not be only one function that is a valid norm. Indeed, generalizing the concept of length allows different norms to be defined on R.
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 morphism 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 morphism 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, bijective 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.