Dual norm

In functional analysis, the dual norm is a measure of size for a continuous linear function defined on a normed vector space.
Definition.
Let formula_1 be a normed vector space with norm formula_2 and let formula_3 be the dual space. The dual norm of a continuous linear functional formula_4 belonging to formula_3 is the non-negative real number defined by any of the following equivalent formulas:
formula_6
which holds for all and . The dual of the dual norm is the original norm: we have formula_7 for all . (This need not hold in infinite-dimensional vector spaces.)
The dual of the Euclidean norm is the Euclidean norm, since
The dual of the formula_12-norm is the formula_13-norm:
and the dual of the formula_13-norm is the formula_16-norm.
More generally, Hölder's inequality shows that the dual of the formula_17-norm is the formula_18-norm, where, satisfies formula_19, i.e., formula_20
As another example, consider the formula_21- or spectral norm on formula_22. The associated dual norm is
which turns out to be the sum of the singular values,
where formula_25 This norm is sometimes called the "nuclear" norm.
Examples.
Dual norm for matrices.
The "Frobenius norm" defined by
is self-dual, i.e., its dual norm is formula_27
The "spectral norm", a special case of the "induced norm" when formula_28, 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 formula_31 where formula_32 denote the singular values.
Some basic results about the operator norm.
More generally, let formula_1 and formula_34 be topological vector spaces and let formula_35 be the collection of all bounded linear mappings (or "operators") of formula_1 into formula_34. In the case where formula_1 and formula_34 are normed vector spaces, formula_35 can be given a canonical norm.
A subset of a normed space is bounded if and only if it lies in some multiple of the unit sphere; thus formula_41 for every formula_42 if formula_43 is a scalar, then formula_44 so that 
The triangle inequality in formula_34 shows that
for every formula_48 satisfying formula_49 This fact together with the definition of formula_50 implies the triangle inequality:
Since formula_52 is a non-empty set of non-negative real numbers, formula_53 is a non-negative real number. 
If formula_54 then formula_55 for some formula_56 which implies that formula_57 and consequently formula_58 This shows that formula_59 is a normed space.
Assume now that formula_34 is complete and we will show that formula_59 is complete. Let formula_62 be a Cauchy sequence in formula_63 so by definition formula_64 as formula_65 This fact together with the relation
implies that formula_67 is a Cauchy sequence in formula_34 for every formula_69 It follows that for every formula_70 the limit formula_71 exists in formula_34 and so we will denote this (necessarily unique) limit by formula_73 that is:
It can be shown that formula_75 is linear. If formula_76, then formula_77 for all sufficiently large integers and . It follows that
for sufficiently all large . Hence formula_79 so that formula_80 and formula_81 This shows that formula_82 in the norm topology of formula_83 This establishes the completeness of formula_83
When formula_34 is a scalar field (i.e. formula_86 or formula_87) so that formula_35 is the dual space formula_3 of formula_1.
Let formula_91denote the closed unit ball of a normed space formula_92 
When formula_34 is the scalar field then formula_94 so part (a) is a corollary of Theorem 1. Fix formula_69 There exists formula_96 such that
but,
for every formula_99. (b) follows from the above. Since the open unit ball formula_100 of formula_1 is dense in formula_31, the definition of formula_103 shows that formula_99 if and only if formula_105 for every formula_106. The proof for (c) now follows directly.