Two theorems are proved. They are with principal significance in functional analysis, for they imply some well-known theorems, such as the open mapping theorem, the closed graph theorem and the Banach-Steinhaus
Schrder studied the method of successive iteration by means of a generalised distance concept. The aim of thise note is to show that in important cases the theorem of J. Schrder can be reformulated as a statement on the convergence of the method of successive approximation in a Banach space...
Theorem 1.1 Hadwiger [Had57] A functional is a continuous, translation and rotation invariant valuation if and only if there are constants such that for every . Hadwiger’s theorem leads to effortless proofs of numerous results in integral geometry and geometric probability (see [Had57] and [...
P. SARKAR Two main ingredients of the proof of these theorems in [3] are analogues of Hausdorff-Young inequality (for radial and K-finite functions proved in =-=[7, 8]-=-) and restriction theorem (proved in [13, 15]); precisely, for 1 ≤ p < 2 and |η| < (2/p − 1)ρ...
electrolysis ⚠️— A tool for formally verifying Rust programs by transpiling them into definitions in the Lean theorem prover. herbie ⚠️— Adds warnings or errors to your crate when using a numerically unstable floating point expression. kani— The Kani Rust Verifier is a bit-precise ...
Elements of Functional Analysis Hindustan Publs. Corp, Delhi, India (1961) Google Scholar 18. G. Mackey Note on a theorem of Murray Bull. Am. Math. Soc., 52 (1946), pp. 322-325 Crossref View in Scopus Google Scholar 19. F.J. Murrary ...
However, a less known but powerful result is that a NN with a single hidden layer can accurately approximate any nonlinear continuous operator. This universal approximation theorem of operators is suggestive of the structure and potential of deep neural networks (DNNs) in learning continuous operators...
Letσ> 1 and letMbe a complete Riemannian manifold. In a very recent work (Grigor′yan and Sun2014), Grigor′yan and Sun proved that a Liouville type theorem holds for nonnegative solutions of elliptic inequality $$ {\Delta} u(x)+u^{\sigma}(x)\leqslant 0,\qquad x\in M. $$ ...
The book covers the standard material in functional analysis, with emphasis on the spectral theory of Hilbert space operators. The basic results on Banach spaces are treated in the first ten lectures: Banach spaces, new spaces from old ones (direct sums, quotients), the Hahn-Banach theorem, ...
(Theorem 3.9 of [15]) LetG,Sbe Abelian groups and suppose thatGis divisible andSis torsion free. Letn\in {\mathbb {N}}be a non-negative integer and let\varphi _{i}, \psi _{i}be homomorphisms ofGonto itself such that Functionsf_{i}:G\rightarrow S \; (i=0, 1, \ldots , n...