# Dual norm

In functional analysis, the **dual norm** is a measure of the "size" of continuous linear functionals defined on a normed space.

## Contents

## Definition[edit]

Let and be topological vector spaces, and ^{[1]} 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 .^{[2]}

**Proof**:

- 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.
^{[3]}

- if is a scalar, then so that
- Assume now that is complete, and that is a Cauchy sequence in .
- Since
- and it is assumed that as n and m tend to , is a Cauchy sequence in for every .
- Hence
- 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
^{[4]}

- Since

**Theorem 2**: Now suppose is the closed unit ball of normed space . Define

for every

- (a) This norm makes into a Banach space.
^{[5]} - (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
^{[6]}such that - but,
- 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)
^{[7]}now follows directly.^{[8]}

- (a) This norm makes into a Banach space.

### The second dual of a Banach space is an isometric isomorphism[edit]

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

and

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 .*

^{[9]}

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.^{[10]}

### Mathematical Optimization[edit]

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.^{[11]} 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.^{[12]}

## Examples[edit]

### Dual norm for matrices[edit]

- 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
^{[citation needed]}.

## See also[edit]

## Notes[edit]

**^**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

## References[edit]

- 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.