where denotes the supremum.
The map defines a norm on . (See Theorems 1 and 2 below.)
The dual norm is a special case of the operator norm defined for each (bounded) linear map between normed vector spaces.
The double dual of a normed linear space
The double dual (or second dual) of is the dual of the normed vector space . There is a natural map . Indeed, for each in define
In general, the map is not surjective. For example, if is the Banach space consisting of bounded functions on the real line with the supremum norm, then the map is not surjective. (See space). If is surjective, then is said to be a reflexive Banach space. If then the space is a reflexive Banach space.
Let be a norm on The associated dual norm, denoted is defined as
(This can be shown to be a norm.) The dual norm can be interpreted as the operator norm of , interpreted as a matrix, with the norm on , and the absolute value on :
From the definition of dual norm we have the inequality
The dual of the Euclidean norm is the Euclidean norm, since
(This follows from the Cauchy–Schwarz inequality; for nonzero z, the value of x that maximises over is .)
The dual of the -norm is the -norm:
and the dual of the -norm is the -norm.
As another example, consider the - or spectral norm on . The associated dual norm is
which turns out to be the sum of the singular values,
where This norm is sometimes called the nuclear norm.
Dual norm for matrices
The Frobenius norm defined by
is self-dual, i.e., its dual norm is
has the nuclear norm as its dual norm, which is defined by
for any matrix where denote the singular values.
Some basic results about the operator norm
More generally, let and be topological vector spaces, and be the collection of all bounded linear mappings (or operators) of into . In the case where and are normed vector spaces, can be normed in a natural way.
- Theorem 1. Let and be normed spaces, and associate to each the number:
- This turns into a normed space. Moreover if is a Banach space, so is .
The triangle inequality in shows that
for every with . Thus
Assume now that is complete, and that is a Cauchy sequence in . Since
and it is assumed that as , is a Cauchy sequence in for every . Hence
exists. It is clear that is linear. If , for sufficiently large n and m. It follows
- Theorem 2. Suppose is the closed unit ball of normed space . For every define:
- (a) This norm makes into a Banach space.
- (b) Let be the closed unit ball of . For every ,
- Consequently, is a bounded linear functional on of norm .
- (c) is weak*-compact.
Proof. Since , when is the scalar field, (a) is a corollary of Theorem 1. Fix . There exists such that
for every . (b) follows from the above. Since the open unit ball of is dense in , the definition of shows that if and only if for every . The proof for (c) now follows directly.
- Rudin 1991, p. 87
- Rudin 1991, section 4.5, p. 95
- Rudin 1991, p. 95
- This inequality is tight, in the following sense: for any x there is a z for which the inequality holds with equality. (Similarly, for any z there is an x that gives equality.)
- Boyd & Vandenberghe 2004, p. 637
- Each is a vector space, with the usual definitions of addition and scalar multiplication of functions; this only depends on the vector space structure of , not .
- Rudin 1991, p. 92
- Rudin 1991, p. 93
- Rudin 1991, p. 93
- Aliprantis 2005, p. 230
- Rudin 1991, Theorem 3.3 Corollary, p. 59
- Rudin 1991, Theorem 3.15 The Banach–Alaoglu theorem algorithm, p. 68
- Rudin 1991, p. 94
- Aliprantis, Charalambos D.; Border, Kim C. (2007). Infinite Dimensional Analysis: A Hitchhiker's Guide (3rd ed.). Springer. ISBN 9783540326960.
- Boyd, Stephen; Vandenberghe, Lieven (2004). Convex Optimization. Cambridge University Press. ISBN 9780521833783.
- Kolmogorov, A.N.; Fomin, S.V. (1957). Elements of the Theory of Functions and Functional Analysis, Volume 1: Metric and Normed Spaces. Rochester: Graylock Press.
- Rudin, Walter (1991), Functional analysis, McGraw-Hill Science, ISBN 978-0-07-054236-5.