Duality of L ^p spaces

Under duality of L ^p -spaces , short L ^p -Dualität , refers to a number of sets of the mathematical field of functional analysis , dedicated to the dual chambers of L ^p -spaces employ, wherein a real number. The essential statement is that dual spaces of L ^p -spaces are again of this kind, namely L ^q -spaces, where must be. That is, applies in a memorable form . ${\displaystyle 1\leq p<\infty }$ ${\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}=1}$ ${\displaystyle (L^{p})\,'\cong L^{q}}$

The case p> 1

Let it be the so-called conjugate exponent, that is, the number for which applies. This is equivalent to . If there is also a measure space , then one can form the Banach spaces and above the body , where stands for or . As usual, matching functions are identified almost everywhere without further information, in order to avoid awkward spelling and writing about equivalence classes of functions. According to Hölder's inequality, the following applies ${\displaystyle q}$ ${\displaystyle p}$ ${\displaystyle 1<q<\infty }$ ${\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}=1}$ ${\displaystyle q={\tfrac {p}{p-1}}}$ ${\displaystyle (X,{\mathcal {A}},\mu )}$ ${\displaystyle L^{p}(X,{\mathcal {A}},\mu )}$ ${\displaystyle L^{q}(X,{\mathcal {A}},\mu )}$ ${\displaystyle \mathbb {K} }$ ${\displaystyle \mathbb {K} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$

${\displaystyle \left|\int _{X}f(x)g(x)\,\mathrm {d} \mu (x)\right|\leq \|f\|_{p}\|g\|_{q}}$for all , ${\displaystyle f\in L^{p}(X,{\mathcal {A}},\mu ),\,g\in L^{q}(X,{\mathcal {A}},\mu )}$

where the norm on the L ^p space denotes and accordingly . This estimate shows that ${\displaystyle \|\cdot \|_{p}}$ ${\displaystyle \|\cdot \|_{q}}$

${\displaystyle T_{g}\colon L^{p}(X,{\mathcal {A}},\mu )\rightarrow \mathbb {K} ,\quad f\mapsto \int _{X}fg\,\mathrm {d} \mu }$

is a restricted linear functional on , i.e. an element of the dual space , with . With the help of Radon-Nikodým's theorem , one can show that every bounded linear functional is of this form and that the norms are even equal. One therefore has the following proposition: ${\displaystyle L^{p}(X,{\mathcal {A}},\mu )}$ ${\displaystyle L^{p}(X,{\mathcal {A}},\mu )\,'}$ ${\displaystyle \|T_{g}\|\leq \|g\|_{q}}$ ${\displaystyle L^{p}(X,{\mathcal {A}},\mu )}$

Let there be a measure space and . Then the picture is ${\displaystyle (X,{\mathcal {A}},\mu )}$ ${\displaystyle 1<p<\infty }$

${\displaystyle T:L^{q}(X,{\mathcal {A}},\mu )\rightarrow L^{p}(X,{\mathcal {A}},\mu )\,',\quad g\mapsto T_{g},\quad T_{g}(f):=\int _{X}fg\,\mathrm {d} \mu }$

an isometric isomorphism .

Exactly this isomorphism is meant when you write briefly . ${\displaystyle L^{p}(X,{\mathcal {A}},\mu )\,'\cong L^{q}(X,{\mathcal {A}},\mu )}$

Since and yes are in a symmetrical relationship to one another, it follows immediately from this sentence ${\displaystyle p}$ ${\displaystyle q}$

${\displaystyle L^{p}(X,{\mathcal {A}},\mu )\,''\cong L^{q}(X,{\mathcal {A}},\mu )\,'\cong L^{p}(X,{\mathcal {A}},\mu )}$.

If one uses the isomorphisms given in the sentence, one recognizes that this is the canonical embedding of in his dual space . The L ^p spaces are therefore reflexive . ${\displaystyle L^{p}(X,{\mathcal {A}},\mu )}$

The above sentence, which is sometimes not quite correctly quoted as Riesz's sentence , has several fathers. The Hilbert space case , which was already proven in 1907, goes back to M. Fréchet . The unit interval stands for the measurement space [0,1] with Borel's σ-algebra and the Lebesgue measure restricted to [0,1] . The generalization of this result to arbitrary Hilbert spaces is also known as the Fréchet-Riesz representation theorem (or Riesz's representation theorem ). F. Riesz proved the case for three years later . OM Nikodým then generalized this to the case of finite measure spaces. The most general case of any measure space was finally treated by EJ McShane in 1950 . ${\displaystyle L^{2}([0,1])}$ ${\displaystyle L^{p}[0,1]}$ ${\displaystyle 1<p<\infty }$

A very simple special case are the sequence spaces that you get when you take and for the measure . The elements from are written as sequences , where such a sequence stands for the function . For the duality between and we get a sum instead of the above integrals: ${\displaystyle \ell ^{p}}$ ${\displaystyle X=\mathbb {N} }$ ${\displaystyle \mu }$ ${\displaystyle \ell ^{p}}$ ${\displaystyle (a_{n})_{n}}$ ${\displaystyle L^{p}}$ ${\displaystyle \mathbb {N} \rightarrow \mathbb {K} ,\,n\mapsto a_{n}}$ ${\displaystyle \ell ^{p}}$ ${\displaystyle \ell ^{q}}$

${\displaystyle T_{b}((a_{n})_{n})=\sum _{n=1}^{\infty }a_{n}b_{n}}$for everyone and . ${\displaystyle (a_{n})_{n}\in \ell ^{p}}$ ${\displaystyle b=(b_{n})_{n}\in \ell ^{q}}$

This statement can also be proven without the expenditure of measure theory.

The case p = 1

A corresponding theorem about the dual space of L ¹ spaces does not apply in full generality. If one forms the exponent conjugated to 1, one must take. In fact, H. Steinhaus could in 1919 ${\displaystyle q=\infty }$

${\displaystyle L^{1}([0,1])\,'\cong L^{\infty }([0,1])}$

show, where the isometric isomorphism is mediated by the operator analogous to the operator defined above . The additional difficulty lies in the fact that the spaces that occur, apart from trivial exceptions, are no longer reflexive. But the following sentence can still be shown: ${\displaystyle T}$

Let it be a -finite measure space. Then the picture is ${\displaystyle (X,{\mathcal {A}},\mu )}$ ${\displaystyle \sigma }$

${\displaystyle T:L^{\infty }(X,{\mathcal {A}},\mu )\rightarrow L^{1}(X,{\mathcal {A}},\mu )\,',\quad g\mapsto T_{g},\quad T_{g}(f):=\int _{X}fg\mathrm {d} \mu }$

an isometric isomorphism.

The additional requirement that the dimensional space is finite cannot be dispensed with. If one considers, for example, the algebra of those sets that are countable or whose complement is countable, and the measure of counting as a measure , then is the space of all functions that are at most different from zero at countably many places and for which applies. Apparently is defined by a bounded linear functional on . If this were of the form for a , then it would have to be constant equal to 1 on and constant equal to 0 on . But such a function is not - measurable . Therefore, in this example, the isomorphism described in the sentence cannot exist. ${\displaystyle \sigma }$ ${\displaystyle X=\mathbb {R} }$ ${\displaystyle \sigma }$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle \mu }$ ${\displaystyle L^{1}(\mathbb {R} ,{\mathcal {A}},\mu )}$ ${\displaystyle f\colon \mathbb {R} \rightarrow \mathbb {K} }$ ${\displaystyle \textstyle \sum _{x\in \mathbb {R} }|f(x)|<\infty }$ ${\displaystyle \textstyle f\mapsto \sum _{x\geq 0}f(x)}$ ${\displaystyle L^{1}(\mathbb {R} ,{\mathcal {A}},\mu )}$ ${\displaystyle T_{g}}$ ${\displaystyle g\in L^{\infty }(\mathbb {R} ,{\mathcal {A}},\mu )}$ ${\displaystyle g}$ ${\displaystyle [0,\infty )}$ ${\displaystyle (-\infty ,0)}$ ${\displaystyle {\mathcal {A}}}$

There is, however, an important situation that also includes certain non- finite measure spaces, in which one still arrives at a satisfactory result, namely that of the locally compact groups . The following sentence is important in harmonic analysis : ${\displaystyle \sigma }$

There are a locally compact group, the Borel algebra on and a regular Borel measure on . Then ${\displaystyle G}$ ${\displaystyle {\mathcal {B}}}$ ${\displaystyle \sigma }$ ${\displaystyle G}$ ${\displaystyle \mu }$ ${\displaystyle G}$

${\displaystyle T\colon L^{\infty }(G,{\mathcal {B}},\mu )\rightarrow L^{1}(G,{\mathcal {B}},\mu )\,',\quad g\mapsto T_{g},\quad T_{g}(f):=\int _{G}fg\,\mathrm {d} \mu }$

an isometric isomorphism .

The measure is called regular if the following three conditions are met: ${\displaystyle \mu }$

• ${\displaystyle \mu (K)<\infty }$for all compact subsets , ${\displaystyle K\subset G}$
• ${\displaystyle \mu (U)=\sup\{\mu (K);\,K\subset U,K{\mbox{ kompakt}}\}}$for all open subsets , ${\displaystyle U\subset G}$
• ${\displaystyle \mu (B)=\inf\{\mu (U);\,B\subset U\subset G,U{\mbox{ offen}}\}}$for all borel quantities . ${\displaystyle B\in {\mathcal {B}}}$

So the phrase applies in particular to the Haar measure on , that is, you can dual space of the group algebra for non describe -endliche groups by sentence above. ${\displaystyle G}$ ${\displaystyle L^{1}(G)}$ ${\displaystyle \sigma }$

The case 0 <p <1

For is L ^p (X, A, μ) not a normed space , but at least a complete topological vector space with the quasi-standard ${\displaystyle 0<p<1}$

${\displaystyle N_{p}\,:\,L^{p}\left(X,{\mathcal {A}},\mu \right)\,\rightarrow \,\mathbb {R} \;,\qquad N_{p}\left(f\right):=\left(\int _{X}\left|f\right|^{p}\mathrm {d} \mu \right)^{\frac {1}{p}}}$

or the pseudo norm or Fréchet metric

${\displaystyle \varrho _{p}\,:\,L^{p}\left(X,{\mathcal {A}},\mu \right)\,\rightarrow \,\mathbb {R} \;,\qquad \varrho _{p}\left(f\right):=\left(N_{p}\left(f\right)\right)^{p}=\int _{X}\left|f\right|^{p}\mathrm {d} \mu \;.}$

These spaces are generally not locally convex , so Hahn-Banach's theorem is generally not applicable, so that there are possibly “very few” linear continuous functionals. In particular, it is not certain that the weak topology can separate into points . ${\displaystyle L^{p}\left(X,{\mathcal {A}},\mu \right)}$

The example with the Borel algebra over the interval and the Borel-Lebesgue measure is prototypical . Here the only convex open sets are the empty set and the entire space itself. Since archetypes of convex open sets in under a linear continuous functional are convex open sets in , it follows that the null functional is the only linear continuous functional. The dual space is therefore trivial: ${\displaystyle L^{p}\left(\left[0,\,1\right]\right):=L^{p}\left(\left[0,\,1\right],{\mathcal {B}}\left(\left[0,\,1\right]\right),\lambda \right)}$ ${\displaystyle {\mathcal {B}}\left(\left[0,\,1\right]\right)}$ ${\displaystyle \left[0,\,1\right]}$ ${\displaystyle \lambda }$ ${\displaystyle \emptyset }$ ${\displaystyle L^{p}\left(\left[0,\,1\right]\right)}$ ${\displaystyle \mathbb {K} }$ ${\displaystyle L^{p}\left(\left[0,\,1\right]\right)}$

${\displaystyle \left(L^{p}\left(\left[0,\,1\right]\right)\right)'=\left\{0\right\}}$.

In particular, the statement of the separation theorem is not valid in this space , since no two points can be separated by a closed hyperplane . The weak topology on is indiscreet . ${\displaystyle L^{p}\left(\left[0,\,1\right]\right)}$

But there are also less extreme examples, such as the sequence spaces with the counting measure . These spaces have nontrivial absolutely convex open sets, but not enough to form a zero neighborhood basis : Since every convex open set in is unbounded, they are not locally convex either. Nevertheless there are “many” linear continuous functionals. It applies to : ${\displaystyle \ell ^{p}:=L^{p}\left(\mathbb {N} ,{\mathcal {P}}\left(\mathbb {N} \right),\mu \right)}$ ${\displaystyle \mu }$ ${\displaystyle \ell ^{p}}$ ${\displaystyle \ell ^{p}}$ ${\displaystyle 0<p<1}$

${\displaystyle \left(\ell ^{p}\right)'=\left(\ell ^{1}\right)'=\ell ^{\infty }\;.}$

The inclusion " " is easy to see because for and applies: ${\displaystyle \supset }$ ${\displaystyle x=\left(x_{k}\right)\in \ell ^{p}}$ ${\displaystyle y=\left(y_{k}\right)\in \ell ^{\infty }}$

${\displaystyle \left|\sum \limits _{k\in \mathbb {N} }x_{k}\,y_{k}\right|\leq \left(\sum \limits _{k\in \mathbb {N} }\left|x_{k}\right|\right)\,\left(\sup \limits _{k\in \mathbb {N} }\left|y_{k}\right|\right)\leq N_{p}\left(x\right)\,\left\|y\right\|_{\infty }\;.}$

For , and the counting measure , i.e. with the quasinorm , the topology in this space is even identical to the usual topology of , since there is exactly one Hausdorff topology on every finite-dimensional real or complex vector space, which turns the space into a topological vector space . Although the spheres in the generating quasinorm are not convex, this generates a locally convex topology: ${\displaystyle X=\left\{1,\,\ldots \,n\right\}}$ ${\displaystyle {\mathcal {A}}={\mathcal {P}}\left(X\right)}$ ${\displaystyle \mu }$ ${\displaystyle L^{p}\left(X,{\mathcal {A}},\mu \right)\cong \mathbb {K} ^{n}}$ ${\displaystyle p}$ ${\displaystyle \mathbb {K} ^{n}}$

${\displaystyle n^{1-{\frac {1}{p}}}\,N_{p}\left(x\right)\leq \left\|x\right\|_{1}\leq N_{p}\left(x\right)\;.}$

The Hahn-Banach theorem is applicable and the dual space again , as in Euclidean or unitary case. The weak topology is identical to the quasinorm topology and the usual topology for the same reasons as above . ${\displaystyle \mathbb {K} ^{n}}$ ${\displaystyle p}$

Banach space-valued L ^p functions

If there is a Banach space in addition to the measure space , then the space of all measurable functions for which the integral is finite can be formed, whereby, as usual, matching functions are identified almost everywhere (see also Bochner integral ). The norm ${\displaystyle (X,{\mathcal {A}},\mu )}$ ${\displaystyle E}$ ${\displaystyle L^{p}(X,{\mathcal {A}},\mu ,E)}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle f\colon X\rightarrow E}$ ${\displaystyle \textstyle \int _{X}\|f(x)\|^{p}\,\mathrm {d} \mu (x)}$

${\displaystyle \|f\|_{p}:=\left(\int _{X}\|f(x)\|^{p}\,\mathrm {d} \mu (x)\right)^{\frac {1}{p}}}$

turns into a Banach space. Are now and , you can ${\displaystyle L^{p}(X,{\mathcal {A}},\mu ,E)}$ ${\displaystyle f\in L^{p}(X,{\mathcal {A}},\mu ,E)}$ ${\displaystyle g\in L^{q}(X,{\mathcal {A}},\mu ,E\,')}$

${\displaystyle \int _{X}gf\,\mathrm {d} \mu =\int _{X}\underbrace {g(x)} _{\in E\,'}(\underbrace {f(x)} _{\in E})\,\mathrm {d} \mu (x)}$

form, and the following applies:

${\displaystyle \left|\int _{X}gf\,\mathrm {d} \mu \right|\leq \int _{X}|g(x)(f(x))|\,\mathrm {d} \mu (x)\leq \int _{X}\|g(x)\|_{q}\|f(x)\|_{p}\,\mathrm {d} \mu (x)\leq \|g\|_{q}\|f\|_{p}}$.

An image is therefore obtained again

${\displaystyle T\colon L^{q}(X,{\mathcal {A}},\mu ,E\,')\rightarrow L^{p}(X,{\mathcal {A}},\mu ,E)\,'}$

and one can show the following theorem:

If a measure space, a separable , reflexive Banach space and as well as the exponent to be conjugated, then is ${\displaystyle (X,{\mathcal {A}},\mu )}$ ${\displaystyle E}$ ${\displaystyle 1<p<\infty }$ ${\displaystyle q}$ ${\displaystyle p}$

${\displaystyle T\colon L^{q}(X,{\mathcal {A}},\mu ,E\,')\rightarrow L^{p}(X,{\mathcal {A}},\mu ,E)\,',\,g\mapsto T_{g},\,T_{g}(f)=\int _{X}gf\,\mathrm {d} \mu }$

an isometric isomorphism.

So the expected and easily memorable formula applies

${\displaystyle L^{p}(X,{\mathcal {A}},\mu ,E)\,'\cong L^{q}(X,{\mathcal {A}},\mu ,E\,')}$.

Weighted l ^p spaces

Let there be a sequence of positive numbers, so-called weights. The associated weighted space is the sequence space ${\displaystyle w=(w_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \ell ^{p}}$

${\displaystyle \ell ^{p}(w):=\left\{(a_{n})_{n}\left|\,\textstyle \sum \limits _{n\in \mathbb {N} }|a_{n}|^{p}w_{n}^{p}<\infty \right.\right\}}$

with the norm

${\displaystyle \|(a_{n})_{n}\|_{p,w}:=\left(\sum _{n\in \mathbb {N} }|a_{n}|^{p}w_{n}^{p}\right)^{\frac {1}{p}}}$.

This is nothing other than space , where the measure is defined by . Applying the above theorem about the L ^p -duality to this, one obtains an isometric isomorphism ${\displaystyle L^{p}(\mathbb {N} ,{\mathcal {P}}(\mathbb {N} ),\mu _{w})}$ ${\displaystyle \mu _{w}}$ ${\displaystyle \mu _{w}(\{n\})=w_{n}}$

${\displaystyle T\colon \ell ^{q}(w)\rightarrow \ell ^{p}(w)\,',\,b=(b_{n})_{n}\mapsto T_{b},\quad T_{b}((a_{n})_{n}):=\sum _{n\in \mathbb {N} }a_{n}b_{n}w_{n}^{p}}$.

In the theory of sequence spaces, one prefers to consider a duality given by the expression , that is, one would like to avoid the factors . To do this, you have to go from sequence to sequence . There , applies ${\displaystyle \textstyle \sum _{n\in \mathbb {N} }a_{n}b_{n}}$ ${\displaystyle w_{n}^{p}}$ ${\displaystyle (b_{n})_{n}\in \ell ^{q}(w)}$ ${\displaystyle (b_{n}w_{n}^{p})_{n}}$ ${\displaystyle p-pq=-q}$

${\displaystyle \|(b_{n})_{n}\|_{q,w}^{q}=\sum _{n\in \mathbb {N} }|b_{n}|^{q}w_{n}^{p}=\sum _{n\in \mathbb {N} }|b_{n}w_{n}^{p}|^{q}w_{n}^{p-pq}=\sum _{n\in \mathbb {N} }|b_{n}w_{n}^{p}|^{q}w_{n}^{-q}=\|(b_{n}w_{n}^{p})_{n}\|_{q,{\frac {1}{w}}}^{q}}$,

where stands for the sequence of weights formed from the reciprocal values . So you get an isometric isomorphism ${\displaystyle {\tfrac {1}{w}}}$ ${\displaystyle w_{n}}$

${\displaystyle \ell ^{q}(w)\rightarrow \ell ^{q}\left({\tfrac {1}{w}}\right),(b_{n})_{n}\mapsto (b_{n}w_{n}^{p})_{n}}$.

If you combine this with the above isometric isomorphism , you get: ${\displaystyle T}$

Let it be a sequence of weights and the exponent to be conjugated. Then ${\displaystyle (w_{n})_{n}}$ ${\displaystyle 1\leq p<\infty }$ ${\displaystyle q}$ ${\displaystyle p}$

${\displaystyle S\colon \ell ^{q}\left({\tfrac {1}{w}}\right)\rightarrow \ell ^{p}(w)\,',\ b=(b_{n})_{n}\mapsto S_{b},\ S_{b}((a_{n})_{n})=\sum _{n\in \mathbb {N} }a_{n}b_{n}}$

an isometric isomorphism.

This isometric isomorphism is what is meant when one

${\displaystyle \ell ^{p}(w)\,'\cong \ell ^{q}\left({\tfrac {1}{w}}\right)}$

writes. It should be pointed out again that this is not the isometric isomorphism from the general theorem about L ^p -duality, except when all weights are equal to 1.

Individual evidence

1. ^ Donald L. Cohn: Measure Theory. Birkhäuser, Boston 1980, ISBN 3-7643-3003-1 , theorem 4.5.17
2. ^ Dunford, Schwartz: Linear Operators, Part I, General Theory. ISBN 0-471-60848-3 , Chapter IV.8, Theorem 1
3. ^ M. Fréchet: Sur les ensembles de fonctions et les opérations linéares , CR Acad Sci Paris 144 (1907), pages 1414-1416
4. ^ F. Riesz: Investigations into systems of integrable functions , Math. Ann. 69 (1910), pp. 449-497
5. ^ OM Nikodým: Contribution à la théorie des fonctionelles linéaires en connexion avec la théorie de la mesure des ensembles abstraits , Mathematica Cluj 5 (1931), pages 130-141
6. ^ EJ McShane: Linear functionals on certain Banach spaces, Proc Amer. Math. Soc. 1 (1950), pages 401-408
7. ↑ H. Steinhaus: Additive and continuous functional operations , Math. Journal 5 (1919), pages 186-221
8. ^ Donald L. Cohn: Measure Theory. Birkhäuser, Boston 1980, ISBN 3-7643-3003-1 , theorem 4.5.17
9. ^ Dunford, Schwartz: Linear Operators, Part I, General Theory. ISBN 0-471-60848-3 , Chapter IV.8, Theorem 5
10. ^ Donald L. Cohn: Measure Theory. Birkhäuser, Boston 1980, ISBN 3-7643-3003-1 , Theorem 9.4.8
11. ↑ ^{a b} Jürgen Elstrodt : Measure and integration theory . 6th edition. Springer Verlag, Berlin, Heidelberg 2009, ISBN 978-3-540-89727-9 , chapter 6, p. 223-225, 229-234, 263, 268 .
12. ^ Herbert Amann, Joachim Escher : Analysis. Volume 3 . 2nd Edition. Birkhäuser Verlag, Basel et al. 2008, ISBN 978-3-7643-8883-6 , Chapter X: Integration Theory, Exercise 13, p. 131 .
13. ^ Walter Rudin : Functional Analysis . 2nd Edition. McGraw-Hill, New York 1991, ISBN 0-07-054236-8 , pp. 36-37 .
14. ^ Hans Wilhelm Alt : Linear functional analysis. An application-oriented introduction . 6th edition. Springer-Verlag, Berlin, Heidelberg 2012, ISBN 978-3-642-22260-3 , Chapter 2. Subsets of function spaces, U2.11, p. 140 .
15. ↑ S. M. Khaleelulla: counterexamples in Topological Vector Spaces . 1st edition. Springer-Verlag, Berlin, Heidelberg 1982, ISBN 978-3-540-39268-2 , Chapter 1 Example 3 (ii), pp. 13 .
16. ↑ Klaus Floret , Joseph Wloka : Introduction to the theory of locally convex spaces . 1st edition. Springer Verlag, Berlin, Heidelberg, New York 1968, ISBN 978-3-540-35855-8 , §3.4, p. 17 .
17. ^ RE Edwards: Functional Analysis: Theory And Applications , Dover Publications, ISBN 0-4866-8143-2 , 8.20
18. ↑ K. Floret, J. Wloka: Introduction to the theory of locally convex spaces , Lecture Notes in Mathematics 56, 1968, ISBN 3-540-04226-1 , §5.4