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| With 2 or 3-dimensional ] with real-valued entries, the idea of the "length" of a vector is intuitive. This can be extended to any ] '''R'''<sup>''n''</sup>. For more abstract ], a '''norm''' is a generalization of this idea. A vector space on which a norm is defined is then called a '''normed vector space'''. | With 2 or 3-dimensional ] with real-valued entries, the idea of the "length" of a vector is intuitive. This can be extended to any ] '''R'''<sup>''n''</sup>. For more abstract ], 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 ] ''K'' (which must be either the ] or the ]), a norm is a ] 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'''||. | For any vector space ''V'' over a ] ''K'' (which must be either the ] or the ]), a norm is a ] 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: | Furthermore, a norm must satisfy the following conditions: | ||
| :For all ''a'' in ''K'' and all '''u''' and '''v''' in ''V'', | :For all ''a'' in ''K'' and all '''u''' and '''v''' in ''V'', | ||
| # ||'''v'''|| ≥ 0, with equality if and only if '''v''' = '''0''' | # ||'''v'''|| ≥ 0, with equality if and only if '''v''' = '''0''' | ||
| # ||''a'''''v'''|| = |''a''|.||'''v'''|| | # ||''a'''''v'''|| = |''a''|.||'''v'''|| | ||
| # ||'''u'''+'''v'''|| ≤ ||'''u'''|| + ||'''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: | 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. | # 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 | # multiplying a vector by a number has the same effect on the length | ||
| ⚫ | # the ], 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 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'''<sup>2</sup> or '''R'''<sup>3</sup>, as well as its generalization to '''R'''<sup>''n''</sup>, 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'''<sup>''n''</sup> instead of '''R'''<sup>''n''</sup> (a normal 'E', not double-barrelled like the 'R'). | The intuitive formula for the length of a vector in '''R'''<sup>2</sup> or '''R'''<sup>3</sup>, as well as its generalization to '''R'''<sup>''n''</sup>, 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'''<sup>''n''</sup> instead of '''R'''<sup>''n''</sup> (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'''<sup>''n''</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>. | ||
| A useful consequence of the norm axioms is the inequality | A useful consequence of the norm axioms is the inequality | ||
| :||'''u''' ± '''v'''|| ≥ | ||'''u'''|| - ||'''v'''|| | | :||'''u''' ± '''v'''|| ≥ | ||'''u'''|| - ||'''v'''|| | | ||
| valid for all vectors '''u''' and '''v'''. | valid for all vectors '''u''' and '''v'''. | ||
| For any normed vector space we can define the ''distance'' between two vectors as ||'''u'''-'''v'''||. | For any normed vector space we can define the ''distance'' between two vectors as ||'''u'''-'''v'''||. | ||
| This makes the normed space into a ] and allows to define notions such as ] and ]. | This makes the normed space into a ] and allows to define notions such as ] and ]. | ||
| If this metric space is ] then the normed space is called a ]. | If this metric space is ] then the normed space is called a ]. | ||
| Every normed vector space sits as a dense subspace inside a Banach space which is called its ''completion''. | Every normed vector space sits as a dense subspace inside a Banach space which is called its ''completion''. | ||
| The most important maps between two normed vector spaces are the ] ]. Together with these maps, normed vector spaces form a ]. An ''isometry'' between two normed vector spaces is a linear map ''f'' which preserves the norm (meaning ||''f''('''v''')|| = ||'''v'''|| for all vectors '''v'''). Isometries are always continuous and ]. A ] isometry between the normed vector spaces ''V'' and ''W'' is called a ''isometric isomorphism'', and ''V'' and ''W'' are called ''isometrically isomorphic''. Isometrically isomorphic normed vector spaces are identical for all practical purposes. | The most important maps between two normed vector spaces are the ] ]. Together with these maps, normed vector spaces form a ]. An ''isometry'' between two normed vector spaces is a linear map ''f'' which preserves the norm (meaning ||''f''('''v''')|| = ||'''v'''|| for all vectors '''v'''). Isometries are always continuous and ]. A ] isometry between the normed vector spaces ''V'' and ''W'' is called a ''isometric isomorphism'', and ''V'' and ''W'' are called ''isometrically isomorphic''. Isometrically isomorphic normed vector spaces are identical for all practical purposes. | ||
| When speaking of normed vector spaces, we augment the notion of dual (see ]) to take the norm into account. 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". The norm of a functional φ is defined as the sup of |φ('''v''')| where '''v''' ranges over all unit vectors (i.e. vectors of norm 1) in ''V''. This turns ''V'' ' into a normed vector space. | When speaking of normed vector spaces, we augment the notion of dual (see ]) to take the norm into account. 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". The norm of a functional φ is defined as the sup of |φ('''v''')| where '''v''' ranges over all unit vectors (i.e. vectors of norm 1) in ''V''. This turns ''V'' ' into a normed vector space. | ||
Revision as of 15:51, 25 February 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 Euclidean 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 generalization 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.
A useful consequence of the norm axioms is the inequality
- ||u ± v|| ≥ | ||u|| - ||v|| |
valid for all vectors u and v.
For any normed vector space we can define the distance between two vectors as ||u-v||. This makes the normed space into a metric space and allows to define notions such as continuity and convergence. If this metric space is complete then the normed space is called a Banach space. Every normed vector space sits as a dense subspace inside a Banach space which is called its completion.
The most important maps between two normed vector spaces are the continuous linear maps. Together with these maps, normed vector spaces form a category. An isometry between two normed vector spaces is a linear map f which preserves the norm (meaning ||f(v)|| = ||v|| for all vectors v). Isometries are always continuous and injective. A surjective isometry between the normed vector spaces V and W is called a isometric isomorphism, and V and W are called isometrically isomorphic. Isometrically isomorphic normed vector spaces are identical for all practical purposes.
When speaking of normed vector spaces, we augment the notion of dual (see dual space) to take the norm into account. 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". The norm of a functional φ is defined as the sup of |φ(v)| where v ranges over all unit vectors (i.e. vectors of norm 1) in V. This turns V ' into a normed vector space.