Quasi-Variational Inequalities in Topological Linear Locally Convex Hausdorff Spaces

1985 ◽  
Vol 122 (1) ◽  
pp. 231-245 ◽  
Author(s):  
Nguyen Xuan Tan
Keyword(s):  
Variational Inequalities ◽  
Hausdorff Spaces ◽  
Topological Linear ◽  
Locally Convex

scholarly journals Notes on differential calculus in topological linear spaces, III

1976 ◽  
Vol 21 (1) ◽  
pp. 88-95
Author(s):  
S. Yamamuro
Keyword(s):  
Real Number ◽  
Number Field ◽  
Open Subset ◽  
Differential Calculus ◽  
Hausdorff Spaces ◽  
Real Numbers ◽  
Linear Spaces ◽  
Topological Linear ◽  
Locally Convex ◽  
Real Number Field

Throughout this note, let E, F and G be locally convex Hausdorff spaces over the real number field R. We denote real numbers by Greek letters. The sets of all continuous semi-norms on E and F will be denoted by P(E) and P(F) respectively, and A will always stand for an open subset of E.

Notes on differential calculus in topological linear spaces, II

1975 ◽  
Vol 20 (2) ◽  
pp. 245-252 ◽  
Author(s):  
S. Yamamuro
Keyword(s):  
Real Number ◽  
Number Field ◽  
Open Subset ◽  
Differential Calculus ◽  
Hausdorff Spaces ◽  
Real Numbers ◽  
Linear Spaces ◽  
Topological Linear ◽  
Locally Convex ◽  
Real Number Field

Throughout this note, let E and F be locally convex Hausdorff spaces over the real number field R. We denote real numbers by Greek letters. The sets of all continuous semi-norms on E and F will be denoted by P(E) and P(F) respectively, and A will always stand for an open subset of E.

scholarly journals A class of nonlinear variational inequalities involving pseudomonotone operators

2000 ◽  
Vol 13 (1) ◽  
pp. 73-75
Author(s):  
Ram U. Verma
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Monotone Operators ◽  
Variational Inequality Problems ◽  
Vector Spaces ◽  
Pseudomonotone Operators ◽  
Topological Vector Spaces ◽  
Locally Convex

We present the solvability of a class of nonlinear variational inequalities involving pseudomonotone operators in a locally convex Hausdorff topological vector spaces setting. The obtained result generalizes similar variational inequality problems on monotone operators.

The graph-theoretic analogue of Tietze's characterization of a convex set and its generalization

1972 ◽  
Vol 72 (1) ◽  
pp. 7-9
Author(s):  
M. D. Guay
Keyword(s):  
Linear Space ◽  
Closed Subset ◽  
Convex Sets ◽  
Convex Set ◽  
Topological Linear Space ◽  
Connected Subset ◽  
Graph Theoretic ◽  
Topological Linear ◽  
Locally Convex

Introduction. One of the most satisfying theorems in the theory of convex sets states that a closed connected subset of a topological linear space which is locally convex is convex. This was first established in En by Tietze and was later extended by other authors (see (3)) to a topological linear space. A generalization of Tietze's theorem which appears in (2) shows if S is a closed subset of a topological linear space such that the set Q of points of local non-convexity of S is of cardinality n < ∞ and S ~ Q is connected, then S is the union of n + 1 or fewer convex sets. (The case n = 0 is Tietze's theorem.)

Statistical convergence in a locally convex space

1988 ◽  
Vol 104 (1) ◽  
pp. 141-145 ◽  
Author(s):  
I. J. Maddox
Keyword(s):  
Convex Space ◽  
Statistical Convergence ◽  
Present Note ◽  
Locally Convex Space ◽  
Strong Summability ◽  
Modulus Function ◽  
Local Convexity ◽  
Complex Sequences ◽  
Topological Linear ◽  
Locally Convex

The notion of statistical convergence was introduced by Fast[1] and has been investigated in a number of papers[2, 5, 6]. Recently, Fridy [2] has shown that k(xk–xk+l) = O(1) is a Tauberian condition for the statistical convergence of (xk). Existing work on statistical convergence appears to have been restricted to real or complex sequences, but in the present note we extend the idea to apply to sequences in any locally convex Hausdorif topological linear space. Also we obtain a representation of statistical convergence in terms of strong summability given by a modulus function, an idea recently introduced in Maddox [3, 4]. Moreover Fridy's Tauberian result is extended so as to apply to sequences of slow oscillation in a locally convex space, and we also examine the local convexity of w(f) spaces.

Precompact linear operators in locally convex spaces

1957 ◽  
Vol 53 (3) ◽  
pp. 581-591 ◽  
Author(s):  
J. R. Ringrose
Keyword(s):  
Hausdorff Space ◽  
General Setting ◽  
Linear Operators ◽  
Fredholm Alternative ◽  
Locally Convex Spaces ◽  
Hausdorff Spaces ◽  
Alternative Theorem ◽  
Image Position ◽  
Complex Number Field ◽  
Locally Convex

This paper is concerned with the properties of precompact and compact linear operators from a locally convex Hausdorff space into itself, the field of scalars being the complex number field.The Riesz–Schauder theory for the equationswhere T is a compact linear operator from the Bacach space into itself, is well known (see, for example, Banach(1), Chapter 10, §2) and has been extended to the more general setting of locally convx Hausdorff spaces by Leray (4). In particular, the ‘Fredholm alternative theorem’ remains valid.

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