bounded linear functionals on
If is a positive measure on a set , , and , where is the Hölder conjugate of , then Hölder’s inequality implies that the map is a bounded linear functional on . It is therefore natural to ask whether or not all such functionals on are of this form for some . Under fairly mild hypotheses, and excepting the case , the Radon-Nikodym Theorem answers this question affirmatively.
If , then the assertion of the theorem remains valid without the assumption that is -finite; however, even with this hypothesis, the result can fail in the case that . In particular, the bounded linear functionals on , where is Lebesgue measure on , are not all obtained in the above manner via members of . An explicit example illustrating this is constructed as follows: the assignment defines a bounded linear functional on , which, by the Hahn-Banach Theorem, may be extended to a bounded linear functional on . Assume for the sake of contradiction that there exists such that for every , and for , define by . As each is continuous, we have for all ; however, because almost everywhere and , the Dominated Convergence Theorem, together with our hypothesis on , gives
a contradiction. It follows that no such can exist.
|Title||bounded linear functionals on|
|Date of creation||2013-03-22 18:32:57|
|Last modified on||2013-03-22 18:32:57|
|Last modified by||azdbacks4234 (14155)|