Partial Differentiation, Differentiation and Continuity on n-Dimensional Real Normed Linear Spaces
نویسندگان
چکیده
منابع مشابه
Partial Differentiation, Differentiation and Continuity on n-Dimensional Real Normed Linear Spaces
The notation and terminology used in this paper have been introduced in the following papers: [2], [7], [1], [3], [4], [5], [17], [11], [13], [6], [9], [14], [10], [8], [12], and [18]. One can prove the following propositions: (1) Let n, i be elements of N, q be an element of Rn, and p be a point of En T. If i ∈ Seg n and q = p, then |pi| ≤ |q|. (2) For every real number x and for every element...
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The notation and terminology used in this paper have been introduced in the following papers: [7], [15], [2], [3], [24], [4], [5], [1], [11], [16], [6], [9], [12], [17], [18], [10], [8], [23], [14], [21], [13], and [22]. For simplicity, we use the following convention: n, m denote non empty elements of N, i, j denote elements of N, f denotes a partial function from 〈Em, ‖ · ‖〉 to 〈En, ‖ · ‖〉, g...
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Let i, n be elements of N. The functor proj(i, n) yielding a function from Rn into R is defined by: (Def. 1) For every element x of Rn holds (proj(i, n))(x) = x(i). Next we state two propositions: (1) dom proj(1, 1) = R1 and rng proj(1, 1) = R and for every element x of R holds (proj(1, 1))(〈x〉) = x and (proj(1, 1))−1(x) = 〈x〉. (2)(i) (proj(1, 1))−1 is a function from R into R1, (ii) (proj(1, 1...
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In this paper i, n, m are elements of N. The following propositions are true: (1) Let f be a set. Then f is a partial function from Rm to Rn if and only if f is a partial function from 〈Em, ‖ · ‖〉 to 〈En, ‖ · ‖〉. (2) Let n, m be non empty elements of N, f be a partial function from Rm to Rn, g be a partial function from 〈Em, ‖·‖〉 to 〈En, ‖·‖〉, x be an element of Rm, and y be a point of 〈Em, ‖ ·...
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The purpose of this paper is to construct a differentiation of operators in Banach spaces. 1,2 are spent to preliminary, in 3,4 derived functions of high order are defined and concretely formulated, in 4.5. generalizations of some theorems in ordinary differentiation will be shown.
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ژورنال
عنوان ژورنال: Formalized Mathematics
سال: 2011
ISSN: 1898-9934,1426-2630
DOI: 10.2478/v10037-011-0011-8