Norm

A mapping from a vector space over the field of real or complex numbers into the real numbers, subject to the conditions:

, and for only;

for every scalar ;

for all (the triangle axiom). The number is called the norm of the element .

A vector space with a distinguished norm is called a normed space. A norm induces on a metric by the formula , hence also a topology compatible with this metric. And so a normed space is endowed with the natural structure of a topological vector space. A normed space that is complete in this metric is called a Banach space. Every normed space has a Banach completion.

A topological vector space is said to be normable if its topology is compatible with some norm. Normability is equivalent to the existence of a convex bounded neighbourhood of zero (a theorem of Kolmogorov, 1934).

The norm in a normed vector space is generated by an inner product (that is, is isometrically isomorphic to a pre-Hilbert space) if and only if for all ,

Two norms and on one and the same vector space are called equivalent if they induce the same topology. This comes to the same thing as the existence of two constants and such that

If is complete in both norms, then their equivalence is a consequence of compatibility. Here compatibility means that the limit relations

imply that .

Not every topological vector space, even if it is assumed to be locally convex, has a continuous norm. For example, there is no continuous norm on an infinite product of straight lines with the topology of coordinate-wise convergence. The absence of a continuous norm can be an obvious obstacle to the continuous imbedding of one topological vector space in another.

If is a closed subspace of a normed space , then the quotient space of cosets by can be endowed with the norm

under which it becomes a normed space. The norm of the image of an element under the quotient mapping is called the quotient norm of with respect to .

The totality of continuous linear functionals on a normed space forms a Banach space relative to the norm

The norms of all functionals are attained at suitable points of the unit ball of the original space if and only if the space is reflexive (cf. Reflexive space).

The totality of continuous (bounded) linear operators from a normed space into a normed space is made into a normed space by introducing the operator norm:

Under this norm is complete if is. When is complete, the space with multiplication (composition) of operators becomes a Banach algebra, since for the operator norm

where is the identity operator (the unit element of the algebra). Other equivalent norms on subject to the same condition are also interesting. Such norms are sometimes called algebraic or ringed. Algebraic norms can be obtained by renorming equivalently and taking the corresponding operator norms; however, even for not all algebraic norms on can be obtained in this manner.

A pre-norm, or semi-norm, on a vector space is defined as a mapping with the properties of a norm except non-degeneracy: does not preclude that . If , a non-zero pre-norm on subject to the condition actually turns out to be a norm (since in this case has no non-trivial two-sided ideals). But for infinite-dimensional normed spaces this is not so. If is a Banach algebra over , then the spectral radius

is a semi-norm if and only if it is uniformly continuous on , and this condition is equivalent to the fact that the quotient algebra by the radical is commutative.

References

[1]	A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)
[2]	W.I. [V.I. Sobolev] Sobolew, "Elemente der Funktionalanalysis" , H. Deutsch , Frankfurt a.M. (1979) (Translated from Russian)
[3]	G.E. Shilov, "Mathematical analysis" , 1–2 , M.I.T. (1974) (Translated from Russian)
[4]	L.V. Kantorovich, G.P. Akilov, "Functionalanalysis in normierten Räumen" , Akademie Verlag (1977) (Translated from Russian)
[5]	W. Rudin, "Functional analysis" , McGraw-Hill (1979)
[6]	M.M. Day, "Normed linear spaces" , Springer (1973)
[7]	I.M. Glazman, Yu.I. Lyubich, "Finite-dimensional linear analysis: a systematic presentation in problem form" , M.I.T. (1974) (Translated from Russian)
[8]	B. Aupetit, "Propriétés spectrales des algèbres de Banach" , Springer (1979)
[9]	A.D. Grishiani, "Theorems and problems in functional analysis" , Springer (1982) (Translated from Russian)

Comments

The theorem that the norms of all functionals are attained at points of the unit ball of the original space if and only if is reflexive is called James' theorem.

References

[a1]	B. Beauzamy, "Introduction to Banach spaces and their geometry" , North-Holland (1982)
[a2]	J. Lindenstrauss, L. Tzafriri, "Classical Banach spaces" , 1–2 , Springer (1977–1979)

For norms in algebra see Norm on a field or ring (see also Valuation).

The norm of a group is the collection of group elements that commute with all subgroups, that is, the intersection of the normalizers of all subgroups (cf. Normalizer of a subset). The norm contains the centre of the group (cf. Centre of a group) and is contained in the second hypercentre . For groups with a trivial centre the norm is the trivial subgroup .

References

[1]	A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian)

O.A. Ivanova

References

[a1]	D.J.S. Robinson, "Finiteness conditions and generalized solvable groups" , 2 , Springer (1972) pp. 45