Highly tempering infinite matrices II: From divergence to convergence via Toeplitz–Silverman matrices

Abstract

It was recently proved [6] that for any Toeplitz-Silverman matrix A, there exists a dense linear subspace of the space of all sequences, all of whose nonzero elements are divergent yet whose images under A are convergent. In this paper, we improve and generalize this result by showing that, under suitable assumptions on the matrix, there are a dense set, a large algebra and a large Banach lattice consisting (except for zero) of such sequences. We show further that one of our hypotheses on the matrix A cannot in general be omitted. The case in which the field of the entries of the matrix is ultrametric is also considered.

Classical vs. non-Archimedean analysis: an approach via algebraic genericity

“…For instance, we can name algebrability and strong algebrability defined in [5,8], respectively. We refer the interested reader to [1,2,[4][5][6][10][11][12][13][14][14][15][16][17][19][20][21][22]24,25,[27][28][29]45] for a current state of the art on this topic.…”

Lineability, spaceability, and latticeability of subsets of C([0, 1]) and Sobolev spaces

“…Furthermore, in 2005 [1], he proved that the set of differentiable nowhere monotone functions also contains (except for {0}) infinite dimensional linear spaces. After these seminal works, a lot has been done linking many areas of mathematics, such as Set Theory [8,9,11], Real and Complex Analysis [10,18], Linear and Multilinear Algebra [5], Linear Dynamics [15], or Statistics [12]. Let us recall some terminology we shall need throughout this work (which can be found in [3,4,22,24]).…”