In functional analysis, the dual norm is a measure of size for a continuous linear function defined on a normed vector space.
The dual norm of a continuous linear functional
is the non-negative real number defined[1] by any of the following equivalent formulas:
map is the origin of the vector space
map and moreover, the sets in the last two rows will both be empty and consequently, their supremums will equal
is not, in general, guaranteed to achieve its norm
is reflexive if and only if every bounded linear function
achieves its norm on the closed unit ball.
[2] It follows, in particular, that every non-reflexive Banach space has some bounded linear functional that does not achieve its norm on the closed unit ball.
However, the Bishop–Phelps theorem guarantees that the set of bounded linear functionals that achieve their norm on the unit sphere of a Banach space is a norm-dense subset of the continuous dual space.
turns out to be stronger than the weak-* topology on
is linear, injective, and distance preserving.
consisting of bounded functions on the real line with the supremum norm, then the map
, is defined by the maximum singular values of a matrix, that is,
denote the singular values[citation needed].
From the definition of dual norm we have the inequality
(This need not hold in infinite-dimensional vector spaces.)
More generally, Hölder's inequality shows that the dual of the
which turns out to be the sum of the singular values,
This inner product can expressed in terms of the norm by using the polarization identity.
which consists of all square-integrable functions, this inner product is
satisfy the polarization identity, and so these dual norms can be used to define inner products.
[10] be the collection of all bounded linear mappings (or operators) of
Assigning to each continuous linear operator
[11] A subset of a normed space is bounded if and only if it lies in some multiple of the unit sphere; thus
is a non-empty set of non-negative real numbers,
and so we will denote this (necessarily unique) limit by
denote the closed unit ball of a normed space
denote the canonical metric induced by the norm on
is a bounded linear functional on a normed space