Dual norm

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In functional analysis, the dual norm is a measure of the "size" of continuous linear functionals.

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:

  1. 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]
  2. 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]

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]

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.

See also[edit]

Notes[edit]

  1. ^ 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 .
  2. ^ Rudin 1991, p. 92
  3. ^ Rudin 1991, p. 93
  4. ^ Rudin 1991, p. 93
  5. ^ 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.
  6. ^ Rudin 1991, Theorem 3.3 Corollary, p. 59
  7. ^ Rudin 1991, Theorem 3.15 The Banach–Alaoglu theorem algorithm, p. 68
  8. ^ Rudin 1991, p. 94
  9. ^ Rudin 1991, Theorem 4.5 The second dual of a Banach space, p. 95
  10. ^ Rudin 1991, p. 95
  11. ^ 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.)
  12. ^ Boyd & Vandenberghe 2004, p. 637

References[edit]

External links[edit]