Related papers: A note on the Schur and Phillips lemmas

We study the classes of filters F on N such that the weak and strong F-convergence of sequences in l1 coincide. We study also an analogue of l1 weak sequential completeness theorem for filter convergence.

In this note the following version of Phillips' lemma is proved. The L-projection of an L-embedded space - that is of a Banach space which is complemented in its bidual such that the norm between the two complementary subspaces is additive…

Let A be a Q-linear pseudo-abelian rigid tensor category. A notion of finiteness due to Kimura and (independently) O'Sullivan guarantees that the ideal of numerically trivial endomorphism of an object is nilpotent. We generalize this result…

In this paper, we establish a converse to Schur's theorem for Lie superalgebras \( L \), focusing on cases where the minimal generator number pairs \((p \vert q)\) of \( L/Z(L) \) are considered, and where the superdimension \(…

To each partition $\lambda$ with distinct parts we assign the probability $Q_\lambda(x) P_\lambda(y)/Z$ where $Q_\lambda$ and $P_\lambda$ are the Schur $Q$-functions and $Z$ is a normalization constant. This measure, which we call the…

A uniformly continuously integrable sequence of real-valued measurable functions, defined on some probability space, is relatively compact in the $\sigma(L^1,L^\infty)$ topology. In this paper, we link such a result to weak convergence…

Here is one of the results obtained in this paper: Let $X, Y$ be two convex sets each in a real vector space, let $J:X\times Y\to {\bf R}$ be convex and without global minima in $X$ and concave in $Y$, and let $\Phi:X\to {\bf R}$ be…

The celebrated Johnson-Lindenstrauss lemma states that for all $\varepsilon \in (0,1)$ and finite sets $X \subseteq \mathbb{R}^N$ with $n>1$ elements, there exists a matrix $\Phi \in \mathbb{R}^{m \times N}$ with…

It is shown that $u_k \cdot v_k$ converges weakly to $u\cdot v$ if $u_k\weakto u$ weakly in $L^p$ and $v_k\weakly v$ weakly in $L^q$ with $p, q\in (1,\infty)$, $1/p+1/q=1$, under the additional assumptions that the sequences $\Div u_k$ and…

The Erberlein-Smulian Theorem asserts that for complete normed spaces, that is Banach spaces, a subset is weak compact if and only if it is weak sequentially compact. In this paper it is shown that the completeness of the normed space is…

In this work, we prove a generalization of Quillen's Theorem A to 2-categories equipped with a special set of morphisms which we think of as weak equivalences, providing sufficient conditions for a 2-functor to induce an equivalence on…

The Ricci version of the Schur theorem is shown to hold for a wide class of Finsler metrics. What is more, let $F$ be any (positive definite) Finsler metric such that $\text{Ric} =\rho F^2$ with $\rho\colon M^n\rightarrow\mathbb{R}$ (i.e.,…

We prove a ``general shrinking lemma'' that resembles the Schwarz--Pick--Ahlfors Lemma and its many generalizations, but differs in applying to maps of a finite disk into a disk, rather than requiring the domain of the map to be complete.…

In a general measure space $(X,\mathcal L,\lambda)$, a characterization of weakly null sequences in $L_\infty (X,\mathcal L,\lambda)$ ($u_k \rightharpoonup 0$) in terms of their pointwise behaviour almost everywhere is derived from the…

We prove that the set of matchings with a fixed number of unmatched vertices is Schur-positive with respect to the set of short chords. Two proofs are presented. The first proof applies a new combinatorial criterion for Schur-positivity,…

Let $\mathcal A = \{A_{ij} \}_{i, j \in \mathcal I}$, where $\mathcal I$ is an index set, be a doubly indexed family of matrices, where $A_{ij}$ is $n_i \times n_j$. For each $i \in \mathcal I$, let $\mathcal V_i$ be an $n_i$-dimensional…

Given a bounded Lipschitz domain $\omega\subset\mathbb{R}^{d-1}$ and a lower semicontinuous function $W:\mathbb{R}^N\to\mathbb{R}_+\cup\{+\infty\}$ that vanishes on a finite set and that is bounded from below by a positive constant at…

This survey paper is based on a talk given at the 44th Summer Symposium in Real Analysis in Paris. This line of research was initiated by a question of Haight and Weizs\"aker concerning almost everywhere convergence properties of series of…

The converse of Schur's lemma (or CSL) condition on a module category has been the subject of considerable study in recent years. In this note we extend that work by developing basic properties of module categories in which the CSL…

The existence of unimodular forms with small norms on sequence spaces is crucial in a variety of problems in modern analysis. We prove that the infimum of $\left\Vert A\right\Vert $ over all unimodular $d$-linear (complex or real) forms $A$…