De Branges Type Lemma and Approximation in Weighted Spaces

The purpose of this paper is to give some generalizations of de Branges Lemma for weighted spaces to obtain different approximation theorems in weighted spaces for algebras, vector subspaces or convex cones. We recall that the (original) de Branges Lemma (Proc Am Math Soc 10(5):822–824, 1959) was demonstrated for continuous scalar function on a compact space while, the weighted spaces are classes of continuous scalar functions on a locally compact space (e.g. the space of function with compact support, the space of bounded functions, the space of functions vanishing at infinity, the space of functions rapidly decreasing at infinity).

Derivations of a Sullivan Model and the Rationalized G -Sequence

Let G k , n ℂ for 2 ≤ k < n denote the Grassmann manifold of k -dimensional vector subspaces of ℂ n . In this paper, we compute, in terms of the Sullivan models, the rational evaluation subgroups and, more generally, the G -sequence of the inclusion G 2 , n ℂ ↣ G 2 , n + r ℂ for r ≥ 1 .

On Forbidden Poset Problems in the Linear Lattice

In this note, we determine the maximum size of a $\{\mathrm{V}_{k}, \Lambda_{l}\}$-free family in the lattice of vector subspaces of a finite vector space both in the non-induced case as well as the induced case, for a large range of parameters $k$ and $l$. These results generalize earlier work by Shahriari and Yu. We also prove a general LYM-type lemma for the linear lattice which resolves a conjecture of Shahriari and Yu.

Rational homotopy of maps between certain complex Grassmann manifolds

LetGn,kdenote the complex Grassmann manifold ofk-dimensional vector subspaces of ℂn. Assumel,k≤ ⌊n/2⌋. We show that, for sufficiently largen, any continuous maph:Gn,l→Gn,kis rationally null homotopic if (i) 1 ≤k<l, (ii) 2 < l <k< 2(l− 1), (iii) 1 < l <k,ldividesnbutldoes not dividek.