Topic
Continuous linear operator
About: Continuous linear operator is a research topic. Over the lifetime, 1826 publications have been published within this topic receiving 36249 citations. The topic is also known as: continuous operator.
Papers published on a yearly basis
1 / 2
Papers
1 Feb 1978
TL;DR: In this article, the authors give an answer to Ulam's problem: "Give conditions in order for a linear mapping near an approximately linear mapping to exist", and prove it for the case n = 1.
Abstract: Let E1, E2 be two Banach spaces, and let f: E1 -* E2 be a mapping, that is "approximately linear". S. M. Ulam posed the problem: "Give conditions in order for a linear mapping near an approximately linear mapping to exist". The purpose of this paper is to give an answer to Ulam's problem. THEOREM. Consider E1, E2 to be two Banach spaces, and let f: E1 -> E2 be a mapping such that f (tx) is continuous in t for each fixed x. Assume that there exists 0 > 0 andp E [0, 1) such that IIf(x + y) f (x) f(A)lI 0. The verification of (3) follows by induction on n. Indeed the case n = 1 is clear because by the hypothesis we can find 0, that is greater or equal to zero, andp such that 0 < p < 1 with 11[f(2x)]/2 -f(x)ll (4) IIxIIp Assume now that (3) holds and we want to prove it for the case (n + 1). However this is true because by (3) we obtain II [f (2n 2x)]/2 n f(2x)llI nI I*2x)2P < m E 2m(P therefore Received by the editors December 1, 1977. AMS (MOS) subject classifications (1970). Primary 47H15; Secondary 39A15.
2,979 citations
1,579 citations
[...]
TL;DR: A(D) as discussed by the authors is a function space with norm ∥ ∥ [Definition I, 3, 1] which defines the topology of major interest in the space; a neighborhood basis of a point x is the family of sets {y: ∥ x - y ∥ ≦ e}.
Abstract: Each of the function spaces mentioned in the introduction of th preceding chapter has (with one exception, A(D)) a norm ∥ ∥ [Definition I, 3, 1] which defines the topology of major interest in the space; a neighborhood basis of a point x is the family of sets {y: ∥ x - y ∥ ≦ e} where e > 0.
1,072 citations
TL;DR: This work extends the Koopman operator to controlled dynamical systems and applies the Extended Dynamic Mode Decomposition (EDMD) to compute a finite-dimensional approximation of the operator in such a way that this approximation has the form of a linearcontrolled dynamical system.
1,063 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 2 |
| 2024 | 4 |
| 2023 | 5 |
| 2022 | 8 |
| 2021 | 6 |
| 2020 | 12 |