On spaces with norms of negative and positive order

Peyser, Gideon

doi:10.1090/S0002-9939-1972-0296665-0

On spaces with norms of negative and positive order
HTML articles powered by AMS MathViewer

by Gideon Peyser PDF

Proc. Amer. Math. Soc. 33 (1972), 81-88 Request permission

Abstract:

The two Hilbert spaces ${H_0}$ and ${H_1}$ are defined to be a generating pair if ${H_1}$ is a dense subspace of ${H_0}$ and if the norm of an element in ${H_1}$ is greater than or equal to the norm in ${H_0}$. It is shown that the pair generates a sequence of spaces $\{ {H_k}\}, - \infty < k < \infty$, such that any two spaces of the sequence form again a generating pair. Such a pair is shown to generate, in turn, a subsequence of $\{ {H_k}\}$. Also, representation theorems are derived for bounded linear functionals over the spaces of the sequence $\{ {H_k}\}$, generalizing the Lax representation theorem and the Lax-Milgram theorem.

References

Edward M. Landesman, Hilbert-space methods in elliptic partial differential equations, Pacific J. Math. 21 (1967), 113–131. MR 209911
Edward M. Landesman, A generalized Lax-Milgram theorem, Proc. Amer. Math. Soc. 19 (1968), 339–344. MR 226375, DOI 10.1090/S0002-9939-1968-0226375-6
Peter D. Lax, On Cauchy’s problem for hyperbolic equations and the differentiability of solutions of elliptic equations, Comm. Pure Appl. Math. 8 (1955), 615–633. MR 78558, DOI 10.1002/cpa.3160080411
P. D. Lax and A. N. Milgram, Parabolic equations, Contributions to the theory of partial differential equations, Annals of Mathematics Studies, no. 33, Princeton University Press, Princeton, N.J., 1954, pp. 167–190. MR 0067317
Louis Nirenberg, Remarks on strongly elliptic partial differential equations, Comm. Pure Appl. Math. 8 (1955), 649–675. MR 75415, DOI 10.1002/cpa.3160080414
Martin Schechter, Negative norms and boundary problems, Ann. of Math. (2) 72 (1960), 581–593. MR 125333, DOI 10.2307/1970230
Kôsaku Yosida, Functional analysis, Die Grundlehren der mathematischen Wissenschaften, Band 123, Academic Press, Inc., New York; Springer-Verlag, Berlin, 1965. MR 0180824