Subspaces Of Finite Codimension Research Articles

On countable codimensional subspaces in Ultra-(DF) spaces

As it is known [3, p. 35] a linear subspace of finite codimension in an Ultra-(DF) space is again an Ultra-(DF) space. In this paper it is proved that an infinite countable codimensional subspace G of an Ultra-(DF) space E is an Ultra-(DF) space, prdvided G has the property (b), i.e. for every bounded subset B of E the codimension of G.in the linear span of GUB is finite. It is shown also that the property of Ultra-(DF) spaces is maintained in subspaces of infinite countable codimension, when the initial Ultra-(DF) space is sequentially complete and bonndedly summing (in the sense of [1]), or an ultrabarrelled space, in particular the Valdivia's result of [9] is obtained.

Subspaces of finite codimension: The existence of elements of best approximation

Subspaces of finite codimension: The existence of elements of best approximation

Remarks on linear selections for the metric projection

The aim of this paper is to give a characterization of the finite-dimensional subspaces of L p , 1 ⩽ p < ∞, and C 0( T) whose metric projections admit linear selections. The paper also gives a characterization of finite co-dimensional subspaces of l 1 and c 0 whose metric projections have linear selections.

Boundary of the unit ball and approximation by elements of subspaces of finite codimension in C(Q)

Boundary of the unit ball and approximation by elements of subspaces of finite codimension in C(Q)

Estimating the distance from a point to a convex set

One estimates the distance between a point and a convex set (cone, subspace) with the help of the distances from the point to the polyhedral sets (polyhedral cones, subspaces of finite codimension) containing the initial one.

Continuity of the metric projection onto subspaces of finite codimension in C(Q)

Continuity of the metric projection onto subspaces of finite codimension in C(Q)

Elements of best approximation relative to subspaces of finite codimension

Elements of best approximation relative to subspaces of finite codimension

Incidence Structures whose Planes are Nets

A d-net is a connected semilinear incidence structure π such that (D1) every plane is a net, (D2) the intersection of two subspaces is connected, (D3) if two planes in a 3-space have a point in common then they have a second point in common, and (D4) the minimum number of points which generate π is d . Let V be a vector space over a skew field F , and W a subspace of finite codimension d . Let P , L be the set of d -, ( d - 1)-dimensional subspaces respectively of V whose intersection with W is the zero vector. The incidence structure ( P , L ⊇ ) is called an attenuated space . We show every d -net for finite d ⩾ 3 is an attenuated space. We also characterize d -nets (together with AG ( d , 2)) as those incidence structures belonging to the diagram where signifies a projective plane and signifies a net.

SUBSPACES OF BARRELLED SPACES

Abstract Dieudone [1] showed that finite-codimensional subspace of a barreled locally convex space is itself barreled, and a simple proof was obtained by Komura [2]. The result was later extended to conutable-condimensional subspaces by Saxon and Levin [3] and Valdivia [4]. Their proofs were rather complicated, and in this note we present a simple proof that a countable-codimensional subspace of a barreled space is barrelled.

Description of the canonical coefficients of the restriction of a quadratic form to a subspace of finite codimension

Description of the canonical coefficients of the restriction of a quadratic form to a subspace of finite codimension

Approximative properties of subspaces of finite codimension in C(Q)

Approximative properties of subspaces of finite codimension in C(Q)

Continuity of metric projection onto a Chebyshev subspace of finite codimension in a space of continuous functions

Continuity of metric projection onto a Chebyshev subspace of finite codimension in a space of continuous functions

Negatively invariant sets of compact maps and an extension of a theorem of Cartwright

Let T be a C 1 map from an open subset of a separable Hilbert space into the Hilbert space, and Γ a negatively invariant compact set, that is T( Γ) ⊇ Γ. Suppose the derivative of T for x ϵ Γ is a uniform contraction on a subspace of finite codimension. Then the topological dimension of Γ is finite. This result may be used to show that for certain delay differential equations and partial differential equations, any almost periodic solution has only finitely many rationally independent frequencies, thus extending results of Cartwright for ODE's.

The continuity of the metric projection on a subspace of finite codimension in the space of continuous functions

The closed subspaces of finite codimension of the space C(X) of all continuous real-valued functions on a compact Hausdorff space X, for which the set of elements of best approximations of every function f e C(X) is nonempty and compact, are characterized. It is shown that if the compact Hausdorff space X is infinite, then C(X) has no subspace of a finite Codimension n > 1 which has a nonempty set of elements of the best approximation for an arbitrary function f 6 e(X) and which has an upper-semicontinuous metric projection.

Finite-Codimensional Subspaces of Countably Quasi-Barrelled Spaces

Finite-Codimensional Subspaces of Countably Quasi-Barrelled Spaces

Finite codimensional subspaces of topological vector spaces and the closed graph theorem

Finite codimensional subspaces of topological vector spaces and the closed graph theorem

Every countable-codimensional subspace of a barrelled space is barrelled

As indicated by the title, the main result of this paper is a straightforward generalization of the following two theorems by J. Dieudonnen and by I. Amemiya and Y. Komura, respectively: (i) Every finite-codimensional subspace of a barrelled space is barrelled. (ii) Every countable-codimensional subspace of a metrizable barrelled space is barrelled. The result strengthens two theorems by G. K6the based on (i) and (ii), and provides examples of spaces satisfying the hypothesis of a theorem by S. Saxon. Introduction. N. Bourbaki [2] observed that if E is a separable, infinite-dimensional Banach space, then E contains a dense subspace M of countably infinite codimension which is a Baire space. R. E. Edwards [4] noted that since M is Baire, it is an example of a noncomplete normed space which is barrelled. Obviously, (i) and (ii) provide a plethora of such examples. It is apparently unknown whether every countable(or even finite-) codimensional subspace of an arbitrary Baire space is Baire; (for closed subspaces the results are affirmative). In the second paper [8], which follows, the authors give topological properties other than barrelledness which are inherited by subspaces having the algebraic property of countable-codimensionality. 1. The notation will be that used by J. Horvath [5]. If (E, F) is a dual pairing (E and F not necessarily separating points) then o(E, F) Presented to the Society, August 30, 1968 under the title On determining barrelled subspaces of barrelled spaces and November 9, 1968; received by the editors December 13, 1968 and, in revised form, June 10, 1970. AMS 1970 subject classifications. Primary 46AO7, 47A55; Secondary 46A30, 46A35, 46A40.

A Note on the Adjoint of the Product of Operators

Cordes and Labrousse ([2] p. 697), and Kaniel and Schechter ([6] p. 429) showed that if S and T are domain-dense closed linear operators on a Hilbert space H into itself, the range of S is closed in H and the codimension of the range of S is finite, then, (TS)* = S*T*. With a somewhat different approach and more restricted condition on S, the same assertion was obtained by Holland [5] recently, that S is a bounded everywhere-defined linear operator whose range is a closed subspace of finite codimension in H.

On properties of subspaces of $l\sb{p},\,0&lt;p&lt;1$

The material presented in this paper deals with some questions con- cerning projections, quotient spaces, and linear dimension in lp spaces, and also includes a remark about weak Schauder bases in lp spaces and an example of an infinite-dimensional closed subspace of /p which is not isomorphic to lp. A great deal is known about the structure of lp spaces for p 2; 1. Perhaps not quite so much is known about these spaces for 0<p< 1. We plan to discuss here proper- ties of subspaces of the latter spaces and show that in most cases these properties are quite different from those of the normed spaces. The paper will be divided into five sections. §1 will contain some comments and definitions which might be helpful to the reader as well as a summary of results. §2 will deal mainly with the concept of complemented subspaces of lp, 0<p<l. We will show in this section that each lp space is isomorphic to all of its subspaces of finite codimension, that each lp space contains a subspace isometrically iso- morphic to lp no infinite-dimensional subspace of which is complemented in /, and that if lp is isometrically isomorphic to one of its subspaces which has the Hahn-Banach extension property, then this subspace is complemented in lp. §3 will contain an example of a subspace of lp which is not isomorphic to lp. §4 will contain some results about subspaces of lp which are obtained as kernels of linear mappings. In particular, we will show that each lp, 0</;<l, contains a closed proper subspace such that any continuous linear functional on lp which vanishes on this subspace vanishes on all of lp. We will also show that lp contains a closed subspace which is not contained in any proper complemented subspace of lp. Finally, §5 will contain some results on linear dimension, complementing those known for lp,p^l. Most of our terminology is standard ; however, a few remarks probably are in order. We use the word norm to denote the lp paranorm when 0<p< I, and we

Characteristic of Chebyshev subspaces of finite codimension in L1

A criterion is established for Chebyshev subspaces of finite codimension in the space of summable functions.