Banach Algebras
... they can be supplied with an extra multiplication operation. A standard example was the space of bounded linear operators on a Banach space, but another important one was function spaces (of continuous, bounded, vanishing at infinity etc. functions as well as functions with absolutely convergent Fou ...
... they can be supplied with an extra multiplication operation. A standard example was the space of bounded linear operators on a Banach space, but another important one was function spaces (of continuous, bounded, vanishing at infinity etc. functions as well as functions with absolutely convergent Fou ...
Diagonal points having dense orbit
... neighborhood U of x. A dynamical system (X, f ) is a minimal system (or f is a minimal map) if the f -orbit of every x ∈ X is dense in X. It is easy to see that (X, f ) is a minimal system iff X has no proper, nonempty, closed f -invariant subset. An element x ∈ X is a minimal point if the restricti ...
... neighborhood U of x. A dynamical system (X, f ) is a minimal system (or f is a minimal map) if the f -orbit of every x ∈ X is dense in X. It is easy to see that (X, f ) is a minimal system iff X has no proper, nonempty, closed f -invariant subset. An element x ∈ X is a minimal point if the restricti ...
Real Analysis - Harvard Mathematics Department
... does exist. The resulting function f (x) however need to be Riemann integrable! To get a reasonable theory that includes such Fourier series, Cantor, Dedekind, Fourier, Lebesgue, etc. were led inexorably to a re-examination of the foundations of real analysis and of mathematics itself. The theory th ...
... does exist. The resulting function f (x) however need to be Riemann integrable! To get a reasonable theory that includes such Fourier series, Cantor, Dedekind, Fourier, Lebesgue, etc. were led inexorably to a re-examination of the foundations of real analysis and of mathematics itself. The theory th ...
The local structure of compactified Jacobians
... The first map is G-invariant and realizes V /H as the quotient of G ×H V by G. The map q is equivariant and can often be described as a contraction onto an orbit. To be precise, suppose that we are given an element v0 ∈ V fixed by H. One may verify that the image of (e, v0 ) ∈ G × V in G ×H V has stab ...
... The first map is G-invariant and realizes V /H as the quotient of G ×H V by G. The map q is equivariant and can often be described as a contraction onto an orbit. To be precise, suppose that we are given an element v0 ∈ V fixed by H. One may verify that the image of (e, v0 ) ∈ G × V in G ×H V has stab ...
Closed sets and the Zariski topology
... The Zariski topology is a coarse topology in the sense that it does not have many open sets. In fact, it turns out that An is what is called a Noetherian space. Definition 1.6. A topological space X is called Noetherian if whenever Y1 ⊃ Y2 ⊃ Y2 ⊃ · · · is a sequence of closed subsets of X, there exi ...
... The Zariski topology is a coarse topology in the sense that it does not have many open sets. In fact, it turns out that An is what is called a Noetherian space. Definition 1.6. A topological space X is called Noetherian if whenever Y1 ⊃ Y2 ⊃ Y2 ⊃ · · · is a sequence of closed subsets of X, there exi ...
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... Definition 2. A function f : X → Y is said to be faintly θ-rare e-continuous at x ∈ X if for each G ∈ θO(Y, f (x)), there exist a θrare set RG in Y with G ∩ Cl(RG ) = ∅ and U ∈ eO(X, x) such that f (U ) ⊂ G ∪ RG . If f is faintly θ-rare e-continuous at every point of X, then it is called faintly θ-r ...
... Definition 2. A function f : X → Y is said to be faintly θ-rare e-continuous at x ∈ X if for each G ∈ θO(Y, f (x)), there exist a θrare set RG in Y with G ∩ Cl(RG ) = ∅ and U ∈ eO(X, x) such that f (U ) ⊂ G ∪ RG . If f is faintly θ-rare e-continuous at every point of X, then it is called faintly θ-r ...
CLASS NOTES ON LINEAR ALGEBRA 1. Matrices Suppose that F is
... dot product on F n . Lemma 1.1. The following are equivalent for two matrices A, B ∈ Mm,n (F ). 1. A = B. 2. Ax = Bx for all x ∈ F n . 3. Aei = Bei for 1 ≤ i ≤ n. N will denote the natural numbers {0, 1, 2 . . .}. ...
... dot product on F n . Lemma 1.1. The following are equivalent for two matrices A, B ∈ Mm,n (F ). 1. A = B. 2. Ax = Bx for all x ∈ F n . 3. Aei = Bei for 1 ≤ i ≤ n. N will denote the natural numbers {0, 1, 2 . . .}. ...
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.