Wikipedia / Dual norm
Wikipedia / 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
be a normed vector space denote its continuous dual space. The dual norm of a continuous linear functional is the non-negative real number defined denote the supremum and infimum map and moreover, the sets in the last two rows will both be empty and consequently, their supremums
). R.C. James proved James's theorem in 1964, which states that a Banach space is reflexive achieves its norm on the closed unit ball. 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.
defines a norm (See Theorems 1 and 2 below.) The dual norm is a special case of the operator norm defined for each (bounded) linear map between normed vector spaces. Since the ground field) is complete is a Banach space induced by turns out to be stronger than the weak-* topology
The double dual of a normed linear space
The double dual
is linear, injective, and distance preserving..
is not surjective. (See L^p space is said to be a reflexive Banach space then the L^p is a reflexive Banach space.
Examples
Dual norm for matrices
The Frobenius norm
The , a special case of the induced norm, is defined by the maximum singular values denote the singular values.
the \ell^p-norm-norm.
Finite-dimensional spaces
The associated dual norm
(This can be shown to be a norm.) The dual norm can be interpreted as the operator norm
(This need not hold in infinite-dimensional vector spaces.)
The dual of the Euclidean norm
(This follows from the Cauchy–Schwarz inequality)
-norm.
More generally, Hölder's inequality shows that the dual of the \ell^p-norm
This norm is sometimes called the nuclear norm.
Lp and ℓp spaces
is some measure space. In particular the Euclidean norm positive definite.
-norm is even induced by a canonical inner product This inner product can expressed in terms of the norm by using the polarization identity this is the associated with a measure space which consists of all square-integrable functions satisfy the polarization identity, and so these dual norms can be used to define inner products. With this inner product, this dual space is also a Hilbert space.
Properties
be the collection of all bounded linear mappings (or 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
is a normed space.
be a Cauchy sequence
for all sufficiently large integers and
is a scalar field is the dual space
is the scalar field There exists if and only if. The proof for (c) now follows directly.
denote the canonical metric denotes the kernel
See also
Notes
References
External links