Orthonormal bases and quasi-splitting subspaces in pre-Hilbert spaces

Let S be a pre-Hilbert space. We study quasi-splitting subspaces of S and compare the class of such subspaces, denoted by Eq(S), with that of splitting subspaces E(S). In [D. Buhagiar, E. Chetcuti, Quasi splitting subspaces in a pre-Hilbert space, Math. Nachr. 280 (5-6) (2007) 479-484] it is proved that if S has a non-zero finite codimension in its completion, then Eq(S)≠E(S). In the present paper it is shown that if S has a total orthonormal system, then Eq(S)=E(S) implies completeness of S. In view of this result, it is natural to study the problem of the existence of a total orthonormal system in a pre-Hilbert space. In particular, it is proved that if every algebraic complement of S in its completion is separable, then S has a total orthonormal system.     

On complete-cocomplete subspaces of an inner product space

In this note we give a measure-theoretic criterion for the completeness of an inner product space. We show that an inner product space S is complete if and only if there exists a -additive state on C(S), the orthomodular poset of complete-cocomplete subspaces of S. We then consider the problem of whether every state on E(S), the class of splitting subspaces of S, can be extended to a Hilbertian state on E( ); we show that for the dense hyperplane S (of a separable Hilbert space) constructed by P. Pták and H. Weber in Proc. Am. Math. Soc. 129 (2001), 2111–2117, every state on E(S) is a restriction of a state on E( ).This revised version was published online in April 2005 with a corrected missing date string.     

A note on the class of paracompact spaces whose product with every paracompact space is paracompact

We prove that if X is a paracompact space which has a neighborhood assignment x→Hx such that for each y∈X the closure of the set is compact then the products T×X, for every paracompact space T, and Xω are paracompact. The first result answers a problem of H. Junnila.     

Homogeneity in powers of subspaces of the real line

Working in ZFC, we prove that for every zero-dimensional subspace of the real line, the Tychonoff power is homogeneous ( denotes the nonnegative integers). It then follows as a corollary that is homogeneous whenever is a separable zero-dimensional metrizable space. The question of homogeneity in powers of this type was first raised by Ben Fitzpatrick, and was subsequently popularized by Gary Gruenhage and Hao-xuan Zhou.

     

Integrability of the general product Hardy operators on the product Hardy spaces

In this paper,we prove that the general product Hardy operators are bounded from the product Hardy space H1/n ( Rm1 ×…× Rmn ) to L 1 ( RΣni=1 mi).     

Sequential completeness of subspaces of products of two cardinals

Let be a cardinal number with the usual order topology. We prove that all subspaces of ² are weakly sequentially complete and, as a corollary, all subspaces of are sequentially complete. Moreover we show that a subspace of (₁ + 1)² need not be sequentially complete, but note that X = A × B is sequentially complete whenever A and B are subspaces of .     

Failure of normality in the box product of uncountably many real lines

We prove in ZFC that the box product of many copies of is neither normal nor collectionwise Hausdorff. As an addendum to the proof, we show that if the cardinality of the continuum is , then these properties also fail in the closed subspace consisting of all functions which assume the value on all but countably many indices.

     

A scalar product for copulas

We introduce a scalar product for n-dimensional copulas, based on the Sobolev scalar product for W1,2-functions. The corresponding norm has quite remarkable properties and provides a new, geometric framework for copulas. We show that, in the bivariate case, it measures invertibility properties of copulas with respect to the ∗-operation introduced by Darsow et al. (1992). The unique copula of minimal norm is the null element for the ∗-operation, whereas the copulas of maximal norm are precisely the invertible elements.     

Inner product spaces and minimal values of functionals

We consider a real function which depends on the distances between a variable point and the points of a finite subset A of a linear normed space X. We show that X is an inner product space if this function attains its local minimum on a barycenter of points of A with well-chosen weights. Our result generalizes classical results about characterization of inner product spaces and answers a question of R. Durier, which was posed in his article [J. Math. Anal. Appl. 207 (1997) 220-239].     

Zero-based subspaces and quasi-invariant subspaces of the Bargmann-Fock space

We use the Weierstrass σ-function associated with a lattice in the complex plane to construct finite dimensional zero-based subspaces and quasi-invariant subspaces of given index in the Bargmann-Fock space.     

Boundedness of Hausdorff operators on some product Hardy type spaces

We study Hausdorff operators on the product Besov space B01,1 (Rn × Rm) and on the local product Hardy space h1 (Rn × Rm).We establish some boundedness criteria for Hausdorff operators on these functio...     

Blaschke products and the rank of backward shift invariant subspaces over the bidisk

Let H²(D²) be the Hardy space over the bidisk. For sequences of Blaschke products {φn(z):−∞<n<∞} and {ψn(w):−∞<n<∞} satisfying some additional conditions, we may define a Rudin type invariant subspace M. We shall determine the rank of H²(D²)?M for the pair of operators and .     

仿紧局部cosmic空间的一些性质

给出仿紧局部cosmic空间的一些特征,建立了仿紧局部cosmic空间的几类序列覆盖L映象的特征,证明了闭L映射和商ss映射保持仿紧局部cosmic空间的性质,此外,还给出仿紧局部cosmic空间的一些映射性质。     

Subsets and subspaces

Several rules known to be valid within the set theory are revisited and considered from the point of view of the question whether they remain satisfied when sets are replaced with subspaces from C_n,1. It is shown that some of the set relationships hold true upon such a replacement, however, there is also an extensive class of relationships whose validity is no longer guaranteed; in this class the commutativity of the orthogonal projectors onto the involved subspaces proves to play an important role. Some related results inspired by probability theory are established as well.     

The topology of ultrafilters as subspaces of 2

Using the property of being completely Baire, countable dense homogeneity and the perfect set property we will be able, under Martin?s Axiom for countable posets, to distinguish non-principal ultrafilters on ω up to homeomorphism. Here, we identify ultrafilters with subpaces of 2^ω in the obvious way. Using the same methods, still under Martin?s Axiom for countable posets, we will construct a non-principal ultrafilter such that is countable dense homogeneous. This consistently answers a question of Hrušák and Zamora Avilés. Finally, we will give some partial results about the relation of such topological properties with the combinatorial property of being a P-point.     

The product of a Baire space with a hereditarily Baire metric space is Baire

In this paper we prove that the product of a Baire space with a metrizable hereditarily Baire space is again a Baire space. This answers a recent question of J. Chaber and R. Pol.

     

On complemented subspaces of sums and products of Banach spaces

It is proved that there exist complemented subspaces of countable topological products (locally convex direct sums) of Banach spaces which cannot be represented as topological products (locally convex direct sums) of Banach spaces.

     

Some remarks on a game of Telgársky

In [3] R. Telgársky (1975) asked: does the first player have a winning strategy in the game G(F,X×X) if the first player has a winning strategy in the game G(F,X)? I give a positive answer to this question and prove that this result is also true for spaces where the first player has a winning strategy in game G(K,X) where K=1, F, C, for σC if X is P-space and for DC if X is collectionwise-normal space. The last result is related to the Telgársky's (1983) conjecture discussed in [1]. These results are not true for infinite product of spaces.     

There is a paracompact Q-set space in ZFC

We construct a paracompact space such that every subset of is an -set, yet is not -discrete. We will construct our space not to have a -diagonal, which answers questions of A.V. Arhangel'skii and D. Shakhmatov on cleavable spaces.