Differentiation in Normed Spaces

Partial Differentiation on Normed Linear Spaces Rn

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...

The Differentiation of Operators in Normed Linear Spaces

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.

-approximation in normed spaces

Differentiation in Normed Spaces1

The notation and terminology used in this paper have been introduced in the following articles: [5], [1], [4], [10], [6], [7], [16], [15], [11], [12], [13], [3], [8], [19], [20], [17], [18], [21], and [9]. Let us consider non empty sets D, E, F . Now we state the propositions: (1) There exists a function I from (FE)D into FD×E such that (i) I is bijective, and (ii) for every function f from D i...

BEST SIMULTANEOUS APPROXIMATION IN FUZZY NORMED SPACES

The main purpose of this paper is to consider the t-best simultaneousapproximation in fuzzy normed spaces. We develop the theory of t-bestsimultaneous approximation in quotient spaces. Then, we discuss the relationshipin t-proximinality and t-Chebyshevity of a given space and its quotientspace.