The Product Space of Real Normed Spaces and its Properties

Abstract

Summary. In this article, we define the product space of real linear spaces and real normed spaces. We also describe properties of these spaces. The Product Space of Real Linear SpacesThe following propositions are true: (1) Let s, t be sequences of real numbers and g be a real number. Suppose that for every element n of N holds t(n) = |s(n) − g|. Then s is convergent and lim s = g if and only if t is convergent and lim t = 0. (2) Let x, y be finite sequences of elements of R. Suppose len x = len y and for every element i of N such that i ∈ Seg len x holds 0 ≤ x(i) and x(i) ≤ y(i). Then |x| ≤ |y|. (3) Let F be a finite sequence of elements of R. If for every element i of N such that i ∈ dom F holds F (i) = 0, then F = 0. 81

Formalization of Orthogonal Decomposition for Hilbert Spaces

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In this article, we formalize the theorems about orthogonal decomposition of Hilbert spaces, using the Mizar system [1], [2]. For any subspace S of a Hilbert space H, any vector can be represented by the sum of a vector in S and a vector orthogonal to S. The formalization of orthogonal complements of Hilbert spaces has been stored in the Mizar Mathematical Library [4]. We referred to [5] and [6] in the formalization.

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The 3-Fold Product Space of Real Normed Spaces and its Properties

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In this article, we formalize in Mizar [1], [2] the 3-fold product space of real normed spaces for usefulness in application fields such as engineering, although the formalization of the 2-fold product space of real normed spaces has been stored in the Mizar Mathematical Library [3].First, we prove some theorems about the 3-variable function and 3-fold Cartesian product for preparation. Then we formalize the definition of 3-fold product space of real linear spaces.

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“…In this article, we formalize in Mizar [1], [2] the 3-fold product space of real normed spaces for usefulness in application fields such as engineering, although the formalization of the 2-fold product space of real normed spaces has been stored in the Mizar Mathematical Library [3].…”

Differentiable Functions on Normed Linear Spaces

“…The notation and terminology used here have been introduced in the following papers: [28], [29], [9], [4], [30], [12], [10], [25], [11], [1], [2], [26], [7], [3], [5], [8], [17], [22], [20], [27], [21], [31], [14], [24], [18], [16], [15], [19], [13], and [6].…”