In
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...
, the dual norm is a measure of size for a
continuous linear function
In mathematics, the term linear function refers to two distinct but related notions:
* In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For dist ...
defined on a
normed vector space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length ...
.
Definition
Let
be a
normed vector space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length ...
with norm
and let
denote its
continuous dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
. The dual norm of a continuous
linear functional
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , the ...
belonging to
is the non-negative real number defined by any of the following equivalent formulas:
where
and
denote the
supremum and infimum, respectively. The constant
map is the origin of the vector space
and it always has norm
If
then the only linear functional on
is the constant
map and moreover, the sets in the last two rows will both be empty and consequently, their
supremums will equal
instead of the correct value of
The map
defines a
norm on
(See Theorems 1 and 2 below.)
The dual norm is a special case of the
operator norm
In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.
Introd ...
defined for each (bounded) linear map between normed vector spaces.
The topology on
induced by turns out to be as strong as the
weak-* topology on
If the
ground field of
is
complete then
is a
Banach space.
The double dual of a normed linear space
The
double dual (or second dual)
of
is the dual of the normed vector space
. There is a natural map
. Indeed, for each
in
define
The map
is
linear,
injective, and
distance preserving. In particular, if
is complete (i.e. a Banach space), then
is an isometry onto a closed subspace of
.
In general, the map
is not surjective. For example, if
is the Banach space
consisting of bounded functions on the real line with the supremum norm, then the map
is not surjective. (See
space). If
is surjective, then
is said to be a
reflexive Banach space. If
then the
space is a reflexive Banach space.
Examples
Dual norm for matrices
The
' defined by
is self-dual, i.e., its dual norm is
The ', a special case of the
''induced norm'' when
, is defined by the maximum
singular values of a matrix, that is,
has the nuclear norm as its dual norm, which is defined by
for any matrix
where
denote the singular values.
If