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Locally Pseudoconvex Spaces and Algebras
Published in Michael Ruzhansky, Hemen Dutta, Advanced Topics in Mathematical Analysis, 2019
Let now (A, τ) be a sequentially advertibly complete Hausdorff LmPA with pseudoconvex von Neumann bornology BA, βA a basis of BA and let B ∈ BA. Then the canonical injection iB from AB into A is continuous (as it was shown above). Therefore, the topology τAB on AB, defined by the system {pnB:n∈ℕ} of seminorms, is stronger than the subspace topology τ|AB on AB. If (an) is an advertibly convergent Cauchy sequence in AB, then there is an element a ∈ AB such that (a ∘ an) and (an ∘ a) converge to zero in the topology τAB. Since it is stronger that τ|AB, then (an) is a Cauchy sequence in A, which advertibly converges in the topology τ as well. Hence, (an) converges in (A, τ), because (A, τ) is sequentially advertibly complete.
Formal power series approach to nonlinear systems with additive static feedback
Published in International Journal of Control, 2023
G. S. Venkatesh, W. Steven Gray
Moreover, the space is not a separable space, viz. the Banach space does not have a countable dense topological subspace (Dahmen et al., 2020). The space , which is a direct product of m Banach spaces, is provided a Banach space structure by the norm Let such that . Observe that as vector spaces. In addition, the topology on induced by the norm is finer than the subspace topology induced from . Hence, this inclusion of vector spaces is not a topological embedding. In fact, the inclusion map is a compact operator, viz. every bounded sequence in has a convergent subsequence in (Dahmen et al., 2020).
Two notions of MV-algebraic semisimplicity relative to fixed MV-chains
Published in Journal of Applied Non-Classical Logics, 2022
Celestin Lele, Jean B. Nganou, Jean M. Wagoum
Conversely, assume that A is -semisimple. Let , which is nonempty subset of . We endow X with the subspace topology of the product topology on . Now, consider the map defined by , where for each . To show that θ is a well-defined MV-algebras homomorphism, we must check that that for each , is a continuous map from . First, note that from the definition of the product topology on , an open subbasis for the topology of X is given by , for and O open in . It follows that for every and every open set O of , , which is open in X.
Coherent measures and the unstable manifold of isolated unstable attractors
Published in Dynamical Systems, 2019
It is clear from the above that if A is an isolated compact invariant set, then A is not necessarily asymptotically stable with respect to the restricted flow in . However, it is possible to define a new topology in , which is finer than the subspace topology, with respect to which the flow remains continuous and A becomes asymptotically stable. Roughly speaking, this new topology is obtained by cutting along the discontinuity set of the final entrance time function with respect to any isolating block of A. It was originally defined in [9]. According to an alternative description given in [11] the intrinsic topology is the smallest topology which contains the subspace topology of and the sets where V runs over the open neighbourhoods of A in M. We denote by the region of attraction equipped with the intrinsic topology. The space is locally compact, separable and metrizable and the topology of A remains unchanged (see [2]).