(1) If a≤ c and d ≤ b, then [c,d]⊆ [a,b]. (2) If a≤ c and b≤ d and c≤ b, then [a,b]∪ [c,d] = [a,d]. (3) If a≤ c and b≤ d and c≤ b, then [a,b]∩ [c,d] = [c,b]. (4) For every subset A of R1 such that A = [a,b] holds A is closed. (5) If a≤ b, then [a, b]T is a closed subspace of R1. (6) If a≤ c and d ≤ b and c≤ d, then [c, d]T is a closed subspace of [a, b]T. (7) If a≤ c and b≤ d and c≤ b, then [a,...

In this paper the existence of minimal lattice-subspaces of a vector lattice E containing a subset B of E+ is studied (a lattice-subspace of E is a subspace of E which is a vector lattice in the induced ordering). It is proved that if there exists a Lebesgue linear topology τ on E and E+ is τ -closed (especially if E is a Banach lattice with order continuous norm), then minimal lattice-subspace...

Let be a regular cardinal. It is proved, among other things, that: (i) if J() is the corresponding long James space, then every closed subspace Y J(), with Dens(Y) = , has a copy of`2 () complemented in J(); (ii) if Y is a closed subspace of the space of continuous functions C((1; ]), with Dens(Y) = , then Y has a copy of c 0 () complemented in C((1; ]). In particular, every nonseparable closed...

a) X is strongly paracompact; b) X is paracompact; c) X is metacompact; d) X is weakly θ-refinable; e) Every open cover of X has a point-countable open refinement; f) X is subparacompact; g) If C is a closed subspace of X, then there are discrete closed sets S and T which are, respectively, well-ordered and reverse-well-ordered by the given ordering of X, have S ∪ T ⊂ C, and have the property t...

This paper provides a subspace method for closed-loop identification, which clearly specifies the model order from noisy measurement data. The can handle long I/O data of target system to be noise-tolerant and determine via nuclear norm minimization. First, proposed compresses by projecting them an appropriate low dimensional subspace, then obtains whose is specified combination compression Its...

Let $\mathcal{W}$ be the wandering subspace corresponding to an invariant of Bergman shift. By identifying space with $H^2(\mathbb{D}^2)\ominus[z-w]$, we give a necessary and sufficient condition for closed $H^2(\mathbb{D}^2)\ominus[z-w]$ subspace. Furthermore, present functional characterization coefficient function in Finally, define operator from one another, get decomposition theorem such o...

Let X be a (real or complex) Banach space, its dual Banach space will be denoted by X *. We use standard notation and terminology of Banach space theory, see J.Lindenstrauss and L.Tzafriri [LT]. By a subspace we mean a linear, but not necessarily closed, subspace. We also assume some knowledge of general topology and ordinal numbers, see P.S.Aleksandrov[A]. Definition 1.1. Let A be a subset of ...