Compact matrix operators on a new sequence space related to ℓp
... For the reader’s convenience, we list a few well-known definitions and results concerning the Hausdorff measure of noncompactness which can be found in [, ], and []. Let S and M be subsets of a metric space (X, d) and ε > . Then S is called an ε-net of M in X if for every x ∈ M there exists s ∈ S ...
... For the reader’s convenience, we list a few well-known definitions and results concerning the Hausdorff measure of noncompactness which can be found in [, ], and []. Let S and M be subsets of a metric space (X, d) and ε > . Then S is called an ε-net of M in X if for every x ∈ M there exists s ∈ S ...
Baire Spaces and the Wijsman Topology
... This theorem was first given by J. Oxtoby in 1957. Then, it was re-discovered by M. R. Krom in 1974, and later on by J. Saint-Raymond in 1983. By a strategy for the player β, we mean a mapping defined for all finite sequences of moves of α. A winning strategy for β is a strategy which can be use ...
... This theorem was first given by J. Oxtoby in 1957. Then, it was re-discovered by M. R. Krom in 1974, and later on by J. Saint-Raymond in 1983. By a strategy for the player β, we mean a mapping defined for all finite sequences of moves of α. A winning strategy for β is a strategy which can be use ...
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... Remark 1.4. Because of the identity αλ = 1, we can express every element of OK in the form a0 + a1 α + a2 α2 + ... + aλ−1 αλ−1 , where a0 , ..., aλ−1 are integers. And because of this form which resembles a polynomial in α, Kummer used the notation f (α), g(α), ... to denote cyclotomic integers. Def ...
... Remark 1.4. Because of the identity αλ = 1, we can express every element of OK in the form a0 + a1 α + a2 α2 + ... + aλ−1 αλ−1 , where a0 , ..., aλ−1 are integers. And because of this form which resembles a polynomial in α, Kummer used the notation f (α), g(α), ... to denote cyclotomic integers. Def ...
Homotopy Theory of Finite Topological Spaces
... with something of a disclaimer, a repudiation of a possible initial fear. What might occur as the homotopy groups of a topological space with only finitely many points? The naive answer would be to assume that these groups must be trivial. This is based, however, on a mistaken intuition, namely that ...
... with something of a disclaimer, a repudiation of a possible initial fear. What might occur as the homotopy groups of a topological space with only finitely many points? The naive answer would be to assume that these groups must be trivial. This is based, however, on a mistaken intuition, namely that ...
EXTERIOR POWERS 1. Introduction Let R be a commutative ring
... f : M k → N is alternating, f vanishes on any k-tuple with a pair of equal coordinates, so fe: M ⊗k → N vanishes on any tensor m1 ⊗ · · · ⊗ mk with mi = mj for some i 6= j. Thus the submodule Jk spanned by these special tensors is automatically in the kernel of fe. The quotient of M ⊗k by the submod ...
... f : M k → N is alternating, f vanishes on any k-tuple with a pair of equal coordinates, so fe: M ⊗k → N vanishes on any tensor m1 ⊗ · · · ⊗ mk with mi = mj for some i 6= j. Thus the submodule Jk spanned by these special tensors is automatically in the kernel of fe. The quotient of M ⊗k by the submod ...
The sequence selection properties of Cp(X)
... Theorem 2.3. For a space X, the following are equivalent: (1) Cp (X) has the weak sequence selection property; (2) X satisfies the following two conditions: (a) IndZ X = 0, (b) for each sequence {Un }n∈ω of clopen γ -covers of X, there is a sequence {Un }n∈ω such that Un ∈ Un (n ∈ ω) and {Un }n∈ω is ...
... Theorem 2.3. For a space X, the following are equivalent: (1) Cp (X) has the weak sequence selection property; (2) X satisfies the following two conditions: (a) IndZ X = 0, (b) for each sequence {Un }n∈ω of clopen γ -covers of X, there is a sequence {Un }n∈ω such that Un ∈ Un (n ∈ ω) and {Un }n∈ω is ...
Three Questions on Special Homeomorphisms on Subgroups of $ R
... topology of XG . To avoid too many variables, we will refer to a topological group under consideration and its underlying topological space by the same letter. For example, if we a topological group G is in our scope and we introduce another group operation ⊕ on the underlying space of G, we will wr ...
... topology of XG . To avoid too many variables, we will refer to a topological group under consideration and its underlying topological space by the same letter. For example, if we a topological group G is in our scope and we introduce another group operation ⊕ on the underlying space of G, we will wr ...
Brouwer fixed-point theorem
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Brouwer. It states that for any continuous function f mapping a compact convex set into itself there is a point x0 such that f(x0) = x0. The simplest forms of Brouwer's theorem are for continuous functions f from a closed interval I in the real numbers to itself or from a closed disk D to itself. A more general form than the latter is for continuous functions from a convex compact subset K of Euclidean space to itself.Among hundreds of fixed-point theorems, Brouwer's is particularly well known, due in part to its use across numerous fields of mathematics.In its original field, this result is one of the key theorems characterizing the topology of Euclidean spaces, along with the Jordan curve theorem, the hairy ball theorem and the Borsuk–Ulam theorem.This gives it a place among the fundamental theorems of topology. The theorem is also used for proving deep results about differential equations and is covered in most introductory courses on differential geometry.It appears in unlikely fields such as game theory. In economics, Brouwer's fixed-point theorem and its extension, the Kakutani fixed-point theorem, play a central role in the proof of existence of general equilibrium in market economies as developed in the 1950s by economics Nobel prize winners Kenneth Arrow and Gérard Debreu.The theorem was first studied in view of work on differential equations by the French mathematicians around Poincaré and Picard.Proving results such as the Poincaré–Bendixson theorem requires the use of topological methods.This work at the end of the 19th century opened into several successive versions of the theorem. The general case was first proved in 1910 by Jacques Hadamard and by Luitzen Egbertus Jan Brouwer.