A mean value theorem for non differentiable mappings in Banach
... It follows from the Borwein-Preiss smooth variational principle [2] that whenever X has an equivalent Fréchet differentiable norm (away from the origin) and f is lower semi continuous, D− f (x) 6= Ø for all x in a dense subset D of X. It is observed in [4] that to get the above result the assumptio ...
... It follows from the Borwein-Preiss smooth variational principle [2] that whenever X has an equivalent Fréchet differentiable norm (away from the origin) and f is lower semi continuous, D− f (x) 6= Ø for all x in a dense subset D of X. It is observed in [4] that to get the above result the assumptio ...
Course 421: Algebraic Topology Section 1
... Proposition 1.2 Let X be a metric space. The collection of open sets in X has the following properties:— (i) the empty set ∅ and the whole set X are both open sets; (ii) the union of any collection of open sets is itself an open set; (iii) the intersection of any finite collection of open sets is i ...
... Proposition 1.2 Let X be a metric space. The collection of open sets in X has the following properties:— (i) the empty set ∅ and the whole set X are both open sets; (ii) the union of any collection of open sets is itself an open set; (iii) the intersection of any finite collection of open sets is i ...
CLOSED GRAPH THEOREMS FOR BORNOLOGICAL
... theorems and also in the hope that in the future bornological vector spaces will gain more popularity and that this work may be useful for others. The closed graph theorem for Banach spaces over R and C is one of the most celebrated classical theorems of functional analysis. Over the years, it has b ...
... theorems and also in the hope that in the future bornological vector spaces will gain more popularity and that this work may be useful for others. The closed graph theorem for Banach spaces over R and C is one of the most celebrated classical theorems of functional analysis. Over the years, it has b ...
Homework #5 Solutions (due 10/10/06)
... Similarly, the subgroups in the eighth row of the table can be identified with the normalizers of cyclic subgroups generated by three-cycles: NS4 (< (123) >) = {e, (123), (132), (12), (13), (23)}, NS4 (< (124) >) = {e, (124), (142), (12), (14), (24)}, NS4 (< (134) >) = {e, (134), (143), (13), (14), ...
... Similarly, the subgroups in the eighth row of the table can be identified with the normalizers of cyclic subgroups generated by three-cycles: NS4 (< (123) >) = {e, (123), (132), (12), (13), (23)}, NS4 (< (124) >) = {e, (124), (142), (12), (14), (24)}, NS4 (< (134) >) = {e, (134), (143), (13), (14), ...
A NOTE ON INVERSE-PRESERVATIONS OF REGULAR OPEN SETS
... f : X ! Y be a Æ-continuous and almost-closed surjection f 1 (y) is N -closed relative to X for each y 2 Y . If X is almost-regular, then so is Y . Proof. Let V be a regular open set of Y and y 2 V . Since f is Æ -continuous, f 1 (V ) is Æ-open in X [8, Theorem 2.2]. For each x 2 f 1 (y) f 1 (V ), ...
... f : X ! Y be a Æ-continuous and almost-closed surjection f 1 (y) is N -closed relative to X for each y 2 Y . If X is almost-regular, then so is Y . Proof. Let V be a regular open set of Y and y 2 V . Since f is Æ -continuous, f 1 (V ) is Æ-open in X [8, Theorem 2.2]. For each x 2 f 1 (y) f 1 (V ), ...
THE CLASSICAL GROUPS
... example is the poset of subspaces, not anything more advanced, such as the structure of a manifold or algebraic variety (though we describe things in enough detail that a more knowledgeable student should easily be able to find such a structure if they know the appropriate defintions). We begin by s ...
... example is the poset of subspaces, not anything more advanced, such as the structure of a manifold or algebraic variety (though we describe things in enough detail that a more knowledgeable student should easily be able to find such a structure if they know the appropriate defintions). We begin by s ...
Covering space
In mathematics, more specifically algebraic topology, a covering map (also covering projection) is a continuous function p from a topological space, C, to a topological space, X, such that each point in X has an open neighbourhood evenly covered by p (as shown in the image); the precise definition is given below. In this case, C is called a covering space and X the base space of the covering projection. The definition implies that every covering map is a local homeomorphism.Covering spaces play an important role in homotopy theory, harmonic analysis, Riemannian geometry and differential topology. In Riemannian geometry for example, ramification is a generalization of the notion of covering maps. Covering spaces are also deeply intertwined with the study of homotopy groups and, in particular, the fundamental group. An important application comes from the result that, if X is a ""sufficiently good"" topological space, there is a bijection between the collection of all isomorphism classes of connected coverings of X and the conjugacy classes of subgroups of the fundamental group of X.