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Sequences and Series of Functions
Published in John D. Ross, Kendall C. Richards, Introductory Analysis, 2020
John D. Ross, Kendall C. Richards
As we have seen, there are two natural definitions of convergence for sequences of functions. Uniform convergence is a stronger condition than pointwise convergence, in the sense that every uniformly convergent sequence of functions is also pointwise convergent, but the converse is not true. A question of interest is which properties of the functions in our sequence carry through the limit under each kind of convergence. A major example is continuity: we have already seen that a sequence of continuous functions (fn)n=1∞ can converge to a discontinuous function f, provided that convergence is pointwise convergence. Surprisingly, this cannot happen in the case of uniform convergence. The following theorem guarantees that the uniform limit of continuous functions is a continuous function.
Global attractors for semigroup actions on uniformizable spaces
Published in Dynamical Systems, 2020
Josiney A. Souza, Richard W. M. Alves
To illustrate problems in the general setting of this paper, we provide examples of semigroup actions on function spaces, which are fundamental components of mathematical analysis. For while, let be the function space of a normed vector space E with the pointwise convergence topology. By considering the multiplicative semigroup of positive integers , let be the action defined as . Then μ defines a semigroup action that is not a classical semiflow and is not metrizable. However, is uniformizable with the uniformity of pointwise convergence. The subspace of all contraction operators with common Lipschitz constant L<1 is μ-invariant. The trivial operator is the global attractor for the restricted action (Example 5.2).
Fixed points for several classes of mappings in variable Lebesgue spaces
Published in Optimization, 2021
T. Domínguez Benavides, S. M. Moshtaghioun, A. Sadeghi Hafshejani
If μ is purely atomic, the weak convergence of a sequence implies the (pointwise) convergence of to zero. (In fact, both types of convergence are equivalent for bounded sequences). By a standard argument, we can construct a subsequence of with almost disjoint support which gives us for an arbitrary and every pair n, m. Thus, for .
Compact almost automorphic solutions for semilinear parabolic evolution equations
Published in Applicable Analysis, 2022
Brahim Es-sebbar, Khalil Ezzinbi, Kamal Khalil
(i) By the pointwise convergence, the function g in Definition 2.2 is only measurable and bounded but not necessarily continuous. If one of the two convergences in Definition 2.2 is uniform on , then f is almost periodic. For more details about this topic we refer the reader to the book [17].