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
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:
![{\displaystyle {\begin{alignedat}{5}\|f\|&=\sup &&\{\,|f(x)|&&~:~\|x\|\leq 1~&&~{\text{ and }}~&&x\in X\}\\&=\sup &&\{\,|f(x)|&&~:~\|x\|<1~&&~{\text{ and }}~&&x\in X\}\\&=\inf &&\{\,c\in [0,\infty )&&~:~|f(x)|\leq c\|x\|~&&~{\text{ for all }}~&&x\in X\}\\&=\sup &&\{\,|f(x)|&&~:~\|x\|=1{\text{ or }}0~&&~{\text{ and }}~&&x\in X\}\\&=\sup &&\{\,|f(x)|&&~:~\|x\|=1~&&~{\text{ and }}~&&x\in X\}\;\;\;{\text{ this equality holds if and only if }}X\neq \{0\}\\&=\sup &&{\bigg \{}\,{\frac {|f(x)|}{\|x\|}}~&&~:~x\neq 0&&~{\text{ and }}~&&x\in X{\bigg \}}\;\;\;{\text{ this equality holds if and only if }}X\neq \{0\}\\\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5616b4447f909e98585faa4c141751a6a30b7462)
where
![{\displaystyle \sup }](https://wikimedia.org/api/rest_v1/media/math/render/svg/266a03117c05ef2558c304b1dbb5f319fcd56fd6)
and
![{\displaystyle \inf }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebda70cf79cae47e82899a2e9dddbd3839f4076c)
denote the supremum and infimum, respectively. The constant
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
map is the origin of the vector space
![{\displaystyle X^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01924e6e5570e2631081fea6c6981b4872d3e04b)
and it always has norm
![{\displaystyle \|0\|=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d6e03564f5aba120ca538006bdbf0cef1070365)
If
![{\displaystyle X=\{0\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bbb567271e86f10b004e7afbb7a348acc7691a0b)
then the only linear functional on
![{\displaystyle X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
is the constant
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
map and moreover, the sets in the last two rows will both be empty and consequently, their supremums will equal
![{\displaystyle \sup \varnothing =-\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb2b29333e7e20077a76d153037ab27bdc6dad3f)
instead of the correct value of
Importantly, a linear function
is not, in general, guaranteed to achieve its norm
on the closed unit ball
meaning that there might not exist any vector
of norm
such that
(if such a vector does exist and if
then
would necessarily have unit norm
). R.C. James proved James's theorem in 1964, which states that a Banach space
is reflexive if and only if every bounded linear function
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.[3][4]
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. Since the ground field of
(
or
) is complete,
is a Banach space. The topology on
induced by
turns out to be stronger than the weak-* topology on
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
![{\displaystyle \varphi (v)(w^{*}):=w^{*}(v).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/041fa81af95e0774a9ac8e1a0de16573d95a4122)
The map
is linear, injective, and distance preserving.[5] In particular, if
is complete (i.e. a Banach space), then
is an isometry onto a closed subspace of
.[6]
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 Frobenius norm defined by
![{\displaystyle \|A\|_{\text{F}}={\sqrt {\sum _{i=1}^{m}\sum _{j=1}^{n}\left|a_{ij}\right|^{2}}}={\sqrt {\operatorname {trace} (A^{*}A)}}={\sqrt {\sum _{i=1}^{\min\{m,n\}}\sigma _{i}^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa1a7bacae8ee0c2848f56b92e11b7c9a0b5addc)
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,
![{\displaystyle \|A\|_{2}=\sigma _{\max }(A),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bfef04129bb71fed00936d2227d041fbcf22c7c)
has the nuclear norm as its dual norm, which is defined by
![{\displaystyle \|B\|'_{2}=\sum _{i}\sigma _{i}(B),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f09464ac2e5e50e3db8c6d0d1cbebf999af9a64d)
for any matrix
![{\displaystyle B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
where
![{\displaystyle \sigma _{i}(B)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d51329d7c33ec6f0dcecac567ca1b478f34f0d)
denote the singular values
[citation needed].
If
the Schatten
-norm on matrices is dual to the Schatten
-norm.
Finite-dimensional spaces
Let
be a norm on
The associated dual norm, denoted
is defined as
![{\displaystyle \|z\|_{*}=\sup\{z^{\intercal }x:\|x\|\leq 1\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22b4e8be3664178bb610272952be964713d3ef95)
(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
:
![{\displaystyle \|z\|_{*}=\sup\{|z^{\intercal }x|:\|x\|\leq 1\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e90495b3a8cfca55f3c5d9400842aee44aca623)
From the definition of dual norm we have the inequality
![{\displaystyle z^{\intercal }x=\|x\|\left(z^{\intercal }{\frac {x}{\|x\|}}\right)\leq \|x\|\|z\|_{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce94dc63a224e4852a80e8b403fc5a14f71bcda6)
which holds for all
![{\displaystyle x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
and
[7] The dual of the dual norm is the original norm: we have
![{\displaystyle \|x\|_{**}=\|x\|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f92e58ad1d2a3cf9fa4882e0862d87747e72555d)
for all
![{\displaystyle x.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d07e9f568a88785ae48006ac3c4b951020f1699a)
(This need not hold in infinite-dimensional vector spaces.)
The dual of the Euclidean norm is the Euclidean norm, since
![{\displaystyle \sup\{z^{\intercal }x:\|x\|_{2}\leq 1\}=\|z\|_{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3573896bd2cc6025a752362c80c95fc624b9eae0)
(This follows from the Cauchy–Schwarz inequality; for nonzero
the value of
that maximises
over
is
)
The dual of the
-norm is the
-norm:
![{\displaystyle \sup\{z^{\intercal }x:\|x\|_{\infty }\leq 1\}=\sum _{i=1}^{n}|z_{i}|=\|z\|_{1},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d1b0d928f8203bf1c83732d80407f4cbfb22ddc)
and the dual of the
![{\displaystyle \ell ^{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d1156e1c2220628042b0fc51e0c73deb3b7c6d1)
-norm is the
![{\displaystyle \ell ^{\infty }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8348195cf09473662c6f59e6717722a6fc01d0f4)
-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
![{\displaystyle \|Z\|_{2*}=\sup\{\mathbf {tr} (Z^{\intercal }X):\|X\|_{2}\leq 1\},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd699f5007746036f59f63d5d3835e152d2bdca2)
which turns out to be the sum of the singular values,
![{\displaystyle \|Z\|_{2*}=\sigma _{1}(Z)+\cdots +\sigma _{r}(Z)=\mathbf {tr} ({\sqrt {Z^{\intercal }Z}}),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9dc5f327ade22187941b799be2df32e4a8317cdf)
where
![{\displaystyle r=\mathbf {rank} Z.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc3e5d9687c9f57c08cf9e8f6eae8063f35d595f)
This norm is sometimes called the
nuclear norm.
[8] Lp and ℓp spaces
For
p-norm (also called
-norm) of vector
is
![{\displaystyle \|\mathbf {x} \|_{p}~:=~\left(\sum _{i=1}^{n}\left|x_{i}\right|^{p}\right)^{1/p}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2fc5b0be78fb024f3b0607bc3387669dcc8816)
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
![{\displaystyle \langle \left(x_{n}\right)_{n},\left(y_{n}\right)_{n}\rangle _{\ell ^{2}}~=~\sum _{n}x_{n}{\overline {y_{n}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed7d653baa4324b70014ec6329e6878a0fa71586)
while for the space
![{\displaystyle L^{2}(X,\mu )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0dd8fa684c40ba3007afc6e6606e1c760307e5dc)
associated with a measure space
![{\displaystyle (X,\Sigma ,\mu ),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28603fd62a4df5a203d25fd1705f67b8a6b82527)
which consists of all
square-integrable functions, this inner product is
![{\displaystyle \langle f,g\rangle _{L^{2}}=\int _{X}f(x){\overline {g(x)}}\,\mathrm {d} x.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41fb0adcb0c7286295d9ad459e547ff7737e0891)
The norms of the continuous dual spaces of
![{\displaystyle \ell ^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91f1f909abd70bd3d8fff0f7ae1ac23052387e18)
and
![{\displaystyle \ell ^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91f1f909abd70bd3d8fff0f7ae1ac23052387e18)
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
Given normed vector spaces
and
let
[9] be the collection of all bounded linear mappings (or operators) of
into
Then
can be given a canonical norm.
Theorem 1 — Let
and
be normed spaces. Assigning to each continuous linear operator
the scalar
![{\displaystyle \|f\|=\sup\{\|f(x)\|:x\in X,\|x\|\leq 1\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ae0e25bb9f866e778d302cdb2a399934fd16be1)
defines a norm
![{\displaystyle \|\cdot \|~:~L(X,Y)\to \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4311d9bdd0a8102c7e7a6c77f54f37f40f796bf)
on
![{\displaystyle L(X,Y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d06d419d8c386f560a7de41b39972a6c55c55141)
that makes
![{\displaystyle L(X,Y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d06d419d8c386f560a7de41b39972a6c55c55141)
into a normed space. Moreover, if
![{\displaystyle Y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f)
is a Banach space then so is
[10] 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 ![{\displaystyle \|\alpha f\|=|\alpha |\|f\|.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1151f3cd07be437dcd0f102787957ee8339467d6) The triangle inequality in shows that ![{\displaystyle {\begin{aligned}\|\left(f_{1}+f_{2}\right)x\|~&=~\|f_{1}x+f_{2}x\|\\&\leq ~\|f_{1}x\|+\|f_{2}x\|\\&\leq ~\left(\|f_{1}\|+\|f_{2}\|\right)\|x\|\\&\leq ~\|f_{1}\|+\|f_{2}\|\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eaac4a780e4a0be37e46a5a30b6d69ed1337d733) for every satisfying This fact together with the definition of implies the triangle inequality: ![{\displaystyle \|f+g\|\leq \|f\|+\|g\|.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77fef7c43d57a499d5b48777a1c1eb6436cbe576) 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.[11] 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 ![{\displaystyle \left\|f_{n}x-f_{m}x\right\|=\left\|\left(f_{n}-f_{m}\right)x\right\|\leq \left\|f_{n}-f_{m}\right\|\|x\|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/529cd372ff65752dbb2e88ad0ba7df1d63f5ee36) 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: ![{\displaystyle fx~=~\lim _{n\to \infty }f_{n}x.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e45098f7ed4b7e1c09121061a06d066d8714199) It can be shown that is linear. If , then for all sufficiently large integers n and m. It follows that ![{\displaystyle \left\|fx-f_{m}x\right\|~\leq ~\varepsilon \|x\|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2bbe7bbbe26c6e91aa18d5bf43146194b8225fe) for sufficiently all large Hence so that and This shows that in the norm topology of This establishes the completeness of [12] |
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
![{\displaystyle \left\|x^{*}\right\|~:=~\sup \left\{|\langle x,x^{*}\rangle |~:~x\in X{\text{ with }}\|x\|\leq 1\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f48c2152bedba40d4849b1ef3f40364459941baf)
where by definition
![{\displaystyle \langle x,x^{*}\rangle ~:=~x^{*}(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/460fdbd045c00885c37e9a95ff2b30201e5e40ad)
is a scalar. Then
is a norm that makes
a Banach space.[13] - If
is the closed unit ball of
then for every
![{\displaystyle {\begin{alignedat}{4}\|x\|~&=~\sup \left\{|\langle x,x^{*}\rangle |~:~x^{*}\in B^{*}\right\}\\&=~\sup \left\{\left|x^{*}(x)\right|~:~\left\|x^{*}\right\|\leq 1{\text{ with }}x^{*}\in X^{*}\right\}.\\\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51a6458e610b6833be9946d16ab91c02d0ab1c92)
Consequently,
is a bounded linear functional on
with norm ![{\displaystyle \|x^{*}\|~=~\|x\|.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a13d1fa28c71b7e095dd8b153f84938511653037)
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[14] such that ![{\displaystyle \langle {x,y^{*}}\rangle =\|x\|.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8776db2a767ad475185c22377c82244a5f5dbf53) but, ![{\displaystyle |\langle {x,x^{*}}\rangle |\leq \|x\|\|x^{*}\|\leq \|x\|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5885d6cc5e0519bb8c6a276abe07fef1f83881f) 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)[15] now follows directly.[16] |
As usual, let
denote the canonical metric induced by the norm on
and denote the distance from a point
to the subset
by
![{\displaystyle d(x,S)~:=~\inf _{s\in S}d(x,s)~=~\inf _{s\in S}\|x-s\|.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/147bb8a98d8fb6c19e44ab738ccba40597336f1e)
If
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
is a bounded linear functional on a normed space
![{\displaystyle X,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df)
then for every vector
![{\displaystyle |f(x)|=\|f\|\,d(x,\ker f),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6231c64434a138d9ff549762132d3e227bb7b159)
where
![{\displaystyle \ker f=\{k\in X:f(k)=0\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e86e290d23448e4dbce7fa022efbbb76886000ca)
denotes the kernel of
See also
Notes
- ^ Rudin 1991, p. 87
- ^ Bishop, Errett; Phelps, R. R. (1961). "A proof that every Banach space is subreflexive". Bulletin of the American Mathematical Society. 67: 97–98. doi:10.1090/s0002-9904-1961-10514-4. MR 0123174.
- ^ Lomonosov, Victor (2000). "A counterexample to the Bishop-Phelps theorem in complex spaces". Israel Journal of Mathematics. 115: 25–28. doi:10.1007/bf02810578. MR 1749671. S2CID 53646715.
- ^ 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 & Border 2006, p. 230
- ^ 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
- Aliprantis, Charalambos D.; Border, Kim C. (2006). Infinite Dimensional Analysis: A Hitchhiker's Guide (3rd ed.). Springer. ISBN 9783540326960.
- Boyd, Stephen; Vandenberghe, Lieven (2004). Convex Optimization. Cambridge University Press. ISBN 9780521833783.
- Diestel, Joe (1984). Sequences and series in Banach spaces. New York: Springer-Verlag. ISBN 0-387-90859-5. OCLC 9556781.
- Hashimoto, Kazuo; Nakamura, Gen; Oharu, Shinnosuke (1986-01-01). "Riesz's lemma and orthogonality in normed spaces" (PDF). Hiroshima Mathematical Journal. 16 (2). Hiroshima University - Department of Mathematics. doi:10.32917/hmj/1206130429. ISSN 0018-2079.
- 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. ISBN 978-1584888666. 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. ISBN 978-0-07-054236-5. 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. ISBN 978-1-4612-7155-0. OCLC 840278135.
- Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
External links
- Notes on the proximal mapping by Lieven Vandenberge
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