Measurement on a normed vector space
In functional analysis, the dual norm is a measure of size for a continuous linear function defined on a normed vector space.
Definition [edit]
Let
be a normed vector space with norm
and let
denote its continuous dual space. The dual norm of a continuous linear functional
belonging to
is the non-negative real number defined[1] 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 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 [edit]
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.[2] In particular, if
is complete (i.e. a Banach space), then
is an isometry onto a closed subspace of
.[3]
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 [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, that is,
has the nuclear norm as its dual norm, which is defined by
for any matrix
where
denote the singular values[ citation needed ].
If
the Schatten
-norm on matrices is dual to the Schatten
-norm.
Finite-dimensional spaces [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
and
[4] The dual of the dual norm is the original norm: we have
for all
(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
the value of
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
satisfies
that is,
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 .[5]
Lp and ℓ p spaces [edit]
For
p-norm (also called
-norm) of vector
is
If
satisfy
then the
and
norms are dual to each other and the same is true of the
and
norms, where
is some measure space. In particular the Euclidean norm is self-dual since
For
, the dual norm is
with
positive definite.
For
the
-norm is even induced by a canonical inner product
meaning that
for all vectors
This inner product can expressed in terms of the norm by using the polarization identity. On
this is the Euclidean inner product defined by
while for the space
associated with a measure space
which consists of all square-integrable functions, this inner product is
The norms of the continuous dual spaces of
and
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 spaces.
Properties [edit]
More generally, let
and
be topological vector spaces and let
[6] be the collection of all bounded linear mappings (or operators) of
into
In the case where
and
are normed vector spaces,
can be given a canonical norm.
Theorem 1 —Let
and
be normed spaces. Assigning to each continuous linear operator
the scalar
defines a norm
on
that makes
into a normed space. Moreover, if
is a Banach space then so is
[7]
Proof |
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 satisfying This fact together with the definition of implies the triangle inequality: Since is a non-empty set of non-negative real numbers, is a non-negative real number. If then for some which implies that and consequently This shows that is a normed space.[8] Assume now that is complete and we will show that is complete. Let be a Cauchy sequence in so by definition as This fact together with the relation implies that is a Cauchy sequence in for every It follows that for every the limit exists in and so we will denote this (necessarily unique) limit by that is: It can be shown that is linear. If , then for all sufficiently large integers n and m. It follows that for sufficiently all large Hence so that and This shows that in the norm topology of This establishes the completeness of [9] |
When
is a scalar field (i.e.
or
) so that
is the dual space
of
.
Theorem 2 —Let
be a normed space and for every
let
where by definition
is a scalar. Then
-
is a norm that makes
a Banach space.[10] - If
is the closed unit ball of
then for every
Consequently,
is a bounded linear functional on
with norm
-
is weak*-compact.
Proof |
Let denote the closed unit ball of a normed space When is the scalar field then so part (a) is a corollary of Theorem 1. Fix There exists[11] 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)[12] now follows directly.[13] |
See also [edit]
- Convex conjugate
- Hölder's inequality – Inequality between integrals in Lp spaces
- Lp space – Function spaces generalizing finite-dimensional p norm spaces
- Operator norm – Measure of the "size" of linear operators
- Polarization identity – Formula relating the norm and the inner product in a inner product space
Notes [edit]
- ^ Rudin 1991, p. 87
- ^ Rudin 1991, section 4.5, p. 95
- ^ Rudin 1991, p. 95
- ^ This inequality is tight, in the following sense: for any
there is a
for which the inequality holds with equality. (Similarly, for any
there is an
that gives equality.) - ^ Boyd & Vandenberghe 2004, p. 637
- ^ 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
. - ^ Rudin 1991, p. 92
- ^ Rudin 1991, p. 93
- ^ Rudin 1991, p. 93
- ^ Aliprantis 2006, p. 230 harvnb error: no target: CITEREFAliprantis2006 (help)
- ^ Rudin 1991, Theorem 3.3 Corollary, p. 59
- ^ Rudin 1991, Theorem 3.15 The Banach–Alaoglu theorem algorithm, p. 68
- ^ Rudin 1991, p. 94
References [edit]
- Aliprantis, Charalambos D.; Border, Kim C. (2006). Infinite Dimensional Analysis: A Hitchhiker's Guide (3rd ed.). Springer. ISBN9783540326960.
- Boyd, Stephen; Vandenberghe, Lieven (2004). Convex Optimization. Cambridge University Press. ISBN9780521833783.
- Kolmogorov, A.N.; Fomin, S.V. (1957). Elements of the Theory of Functions and Functional Analysis, Volume 1: Metric and Normed Spaces. Rochester: Graylock Press.
- Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN978-1584888666. OCLC 144216834.
- Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN978-0-07-054236-5. OCLC 21163277.
- Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN978-1-4612-7155-0. OCLC 840278135.
- Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN978-0-486-45352-1. OCLC 853623322.
External links [edit]
- Notes on the proximal mapping by Lieven Vandenberge
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