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In this article, we define and develop partial differentiation of vector-valued functions on n-dimensional real normed linear spaces (refer to [19] and [20]). The notation and terminology used in this paper have been introduced in the 2 takao inoué et al. (6) Let m, i be elements of N, x be an element of R m , and r be a real number. m))(r − (proj(i,(More)
In this paper, we first introduce the notion of summability of an infinite set of vectors of real Hilbert space, without using index sets. Further we introduce the notion of weak summability, which is weaker than that of summability. Then, several statements for summable sets and weakly summable ones are proved. In the last part of the paper, we give a(More)
In this article, we define and develop differentiation of vector-valued functions on n-dimensional real normed linear spaces (refer to [16] and [17]). 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 R m to R n if and only if f is a partial function from E m , · to E n , ·.(More)
INTRODUCTION Desert dust is estimated to constitute about 35% of aerosol in the troposphere. Desertification, climatic variability and global warming all can contribute to increased dust formation. This study aims to examine possible health effects of desert dust exposure on pregnant women and their children. The purpose of this report was to present the(More)
OBJECTIVES The present study was performed to elucidate the effects of individual and social factors on smoking behavior of parents of fourth grade elementary school students in Japan. METHODS A self-administered questionnaire was sent to a total of 4,179 households of fourth grade elementary school students. A total of 3,522 responses including actual(More)
This paper is a continuation of our paper [21]. We give an analogue of the necessary and sufficient condition for summable set (i.e. the main theorem of [21]) with respect to summable set by a functional L in real Hilbert space. After presenting certain useful lemmas, we prove our main theorem that the summability for an orthonormal infinite set in real(More)