About the Book
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 60. Chapters: Atkinson Mingarelli theorem, ATS theorem, Babu ka Lax Milgram theorem, Balian Low theorem, Bauer Fike theorem, Besicovitch covering theorem, Beurling Lax theorem, Borel's lemma, Browder Minty theorem, Caratheodory's existence theorem, Carleson's theorem, Cartan Kahler theorem, Cartan Kuranishi prolongation theorem, Cauchy formula for repeated integration, Cauchy Kowalevski theorem, Danskin's theorem, Differentiation of integrals, Dirichlet conditions, Envelope theorem, Equioscillation theorem, Euler Maclaurin formula, Faa di Bruno's formula, Fenchel's duality theorem, Final value theorem, Fuchs's theorem, Gaussian integral, Godunov's theorem, Goldbach Euler theorem, Gradient conjecture, Hadamard's lemma, Hardy Littlewood tauberian theorem, Hartman Grobman theorem, Hausdorff paradox, Heine Cantor theorem, Helly's selection theorem, Helmholtz decomposition, Hobby Rice theorem, Holder's theorem, Holmgren's uniqueness theorem, Implicit function, Infinite-dimensional Lebesgue measure, Initial value theorem, Integral representation theorem for classical Wiener space, Kantorovich theorem, Kneser's theorem (differential equations), Lagrange reversion theorem, Laplace principle (large deviations theory), Lax equivalence theorem, Lax Wendroff theorem, Mahler's theorem, Malgrange preparation theorem, Malgrange Ehrenpreis theorem, Mellin inversion theorem, Mountain pass theorem, Peano existence theorem, Picard Lindelof theorem, Poisson summation formula, Pontryagin duality, Prekopa Leindler inequality, Projection-slice theorem, Rademacher Menchov theorem, Rellich Kondrachov theorem, Riemann series theorem, Riemann Lebesgue lemma, Sard's theorem, Shift theorem, Silverman Toeplitz theorem, Stirling's approximation, Sturm separation theorem, Sturm Picone comparison theorem, Symmetry of second derivatives, Szeg limit theorems, Tonelli Hobson test, Trudinger's theorem, Whitney extension theorem, Wiener's tauberian theorem, Wiener Ikehara theorem, Wirtinger's inequality for functions, Zahorski theorem. Excerpt: In mathematics, specifically in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform on locally compact groups, such as R, the circle or finite cyclic groups. The Pontryagin duality theorem itself states that locally compact groups identify naturally with their bidual. The subject is named after Lev Semenovich Pontryagin who laid down the foundations for the theory of locally compact abelian groups and their duality during his early mathematical works in 1934. Pontryagin's treatment relied on the group being second-countable and either compact or discrete. This was improved to cover the general locally compact abelian groups by Egbert van Kampen in 1935 and Andre Weil in 1940. Pontryagin duality places in a unified context a number of observations about functions on the real line or on finite abelian groups: The theory, introduced by Lev Pontryagin and combined with Haar measure introduced by John von Neumann, Andre Weil and others depends on the theory of the dual group of a locally compact abelian group. It is analogous to the dual vector space of a vector space: a finite-dimensional vector space V and its dual vector space V* are not naturally isomorphic, but their endomorphism algebras (matrix algebras) are: End(V) End(V*), via the transpose. Similarly, a group G and its dual group G DEGREES are not in general isomorphic, but their group algebras are: C(G) C(G DEGREES) via the Fourier transform, though one must carefully define these algebras analytically. More categorically, this is not just an isomorphism of endomorphism algebras, but an isomorphism of categories see categorical considerations. A topological group is locally compact if...
