Weighted E-Spaces and Epistemic Information Operators

Abstract

Information is usually related to knowledge. Here, we present a broader picture in which information is associated with epistemic structures, which form cognitive infological systems as basic recipients and creators of cognitive information. Infological systems are modeled by epistemic spaces, while operators in these spaces are mathematical models of information. Information that acts on epistemic structures is called cognitive information, while information that acts on knowledge structures is called epistemic information. The latter brings new and updates existing knowledge, being of primary importance to people. In this paper, both types of information are studied as operators in epistemic spaces based on the general theory of information. As a synthetic approach, which reveals the essence of information, organizing and encompassing all main directions in information theory, the general theory of information provides efficient means for such a study. Different types of information dynamics representation use tools from various mathematical disciplines, such as the theory of categories, functional analysis, mathematical logic and algebra. In this paper, we base our exploration of information and knowledge dynamics on functional analysis further developing the mathematical stratum of the general theory of information.Keywords: information; knowledge; epistemic structure; epistemic space; weighted epistemic space; epistemic information operator; vector bundle; continuity; boundedness definition of the rectangular closure R(X), there are points x and z from X such x = (a, w) and z = (b, y) with a, b ∈ X e. By the properties of metric,By initial conditions, d(x, z) < k. At the same time, by the definition of the metric d and Corollary 2, we have: d((c, w), (a, w)Proposition is proved because p and q are arbitrary points from R(X). □Reducing the problem of boundedness to rectangular sets, now we find conditions of boundedness for rectangular sets.Proposition 3. A rectangular subset X of the space E of the vector bundle E = (E, p E , E e ) is bounded if and only if the projection X e = p E (X) of X and the fiber projection σ(X) of X are uniformly bounded. Proof. Necessity. Let us assume that the projection X e = p E (X) of X is unbounded. It means that for any positive number k, there are two points a and b in X e such that d(a, b) > k. As X e is the projection of X, there are two points x and z in X such that a = p E (x) and b = p E (z). By the definition of the metric in the space E, d(x, z) ≥ d(a, b) > k. Consequently, X is also unbounded. Now, let us suppose that the fiber projection σ(X) of X is not uniformly bounded. It means that for any positive number k, there are two points u and v in σ(X) such that d v (u, v) > k. As σ(X) is a projection of X, there are points x = (a, u) and z = (b, v) from the space X. By Corollary 2, d(x, z) ≥ k as by choice of the points u and v, d v (u, v) > k. Thus, the space X is not bounded.Then by the Law of Contraposition, if the space X is bounded, then the projection X e = p E ...

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