In functional analysis, the dual norm is a measure of the "size" of continuous linear functionals.
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.
When is a scalar field (i.e. or ) so that is the dual space of , the norm on defines a topology on which turns out to be stronger than its weak-*topology.
Theorem 1: Let and be normed spaces, and associate to each the number:
We first establish that is bounded (using the triangle inequality), and complete (using Cauchy sequences) using our definition of , thereby making a normed space. If is a Banach space, so is .
- A subset of a normed space is bounded if and only if it lies in some multiple of the unit sphere; thus for every
- if is a scalar, then so that
- The triangle inequality in shows that
- for every with . Thus
- If , then for some ; hence . Thus, is a normed space.
- Assume now that is complete, and that is a Cauchy sequence in .
- and it is assumed that as n and m tend to , is a Cauchy sequence in for every .
- exists. It is clear that is linear. If , for sufficiently large n and m. It follows
- for sufficiently large m.
- Hence , so that and .
- Thus in the norm of . This establishes the completeness of 
Theorem 2: Now suppose is the closed unit ball of normed space . 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.
- 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.
The second dual of a Banach space is an isometric isomorphism
The normed dual of a Banach space is also a Banach space, which means it has a normed dual, , of its own.
By part (b) of Theorem 2, every defines a unique by equation
It follows from the first and second equation that is linear and is an isometry. Given that is assumed to be complete, is closed in .
Thus, is an isometric isomorphism onto a closed subspace of .
The members of are exactly the linear functionals on that are continuous with respect to its weak*-topology. Since the norm topology of is stronger, may happen that is a proper subspace of .
However, there are many important spaces, such as the Lp spaces with , where ; these are called reflexive.
It is stressed that, for to be reflexive, the existence of some isometric isomorphism of onto is not enough; it is crucial that satisfies first equation in this section.
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
which holds for all x and z. The dual of the dual norm is the original norm: we have for all x. (This need not hold in infinite-dimensional vector spaces.)
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.
More generally, Hölder's inequality shows that the dual of the -norm is the -norm, where, q satisfies , i.e.,
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 .
- The spectral norm, a special case of the induced norm when , is defined by the maximum singular values of a matrix, i.e.,
- has the nuclear norm as its dual norm, which is defined by for any matrix where denote the singular values.
- ^ 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
6.7 Definition The norm dual of a normed space is Banach space . The operator norm on is also called the dual norm, also denoted . That is,
The dual space is indeed a Banach space by Theorem 6.6.
- ^ Rudin 1991, Theorem 3.3 Corollary, p. 59
- ^ Rudin 1991, Theorem 3.15 The Banach–Alaoglu theorem algorithm, p. 68
- ^ Rudin 1991, p. 94
- ^ Rudin 1991, Theorem 4.5 The second dual of a Banach space, 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
- 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.