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In this paper, two new types of orthogonality from the generalized Carlssion orthogonality have been studied and some properties of orthogonality in Banach spaces are verified and as best implies Birkhoff orthogonality and Birkhoff orthogonality implies best approximation, in this paper, Pythagorean orthogonality also implies best approximation has been proved. Keywords:Birkhof orthogonality, James orthogonality, Carlsson orthogonality, Pythagorean orthogonality, Best approximation. IntroductionOrthogonality is one of the most important concepts in the theory of inner product spaces. Let X be an inner product space. Then a vector x ∈ X is said to be orthogonal to the vector y ∈ X if (x, y) = 0. However, when switching from the theory of inner product spaces to normed linear spaces, we cannot maintain the notion of orthogonality. Therefore, since 1934 generalized orthogonality types for normed linear spaces have been introduced and studied. When the norm of X is induced by an inner product, the orthogonality in X is equivalent to each of the following propositions. This motivates that in context of normed linear spaces, any one of the following proposition can be taken as a definition of orthogonality in X.Roberts (

Aims: To compare the effectiveness of mifepristone plus misoprostol in pre-induction cervical ripening and induction of labour in uncomplicated post-dated primigravidae. Methods: It is a hospital based prospective comparative study enrolling 50 uncomplicated post-dated pregnancies in each group to determine the efficacy and safety of mifepristone and misoprostol with misoprostol alone for induction. It was conducted in Institute of Medicine. Computer software SPSS 20 was used for processing and analysis of the data. Chi-square test and paired sample t-test was used. Results: Comparing change in Bishops score among the two groups, no any statistically significant difference was seen. Sixty-six percent delivered vaginally in mifepristone group as compared to 42% in misoprostol group (p≤0.01). Rest delivered by caesarean section. No significant maternal side effects and complications were seen. Twenty-six percent in mifepristone group and 50% babies in misoprostol group developed fetal side effects (p=0.01). Conclusions: Mifepristone and misoprostol both bring favourable change in the Bishop’s score. Regarding success rate and fetal complications mifepristone seems to be effective and safe. Keywords: induction of labour, mifepristone, misoprostol, post-dated pregnancy

In an arbitrary normed space, though the norm not necessarily coming from the inner product space, the notion of orthogonality may be introduced in various ways as suggested by the mathematicians like R.C. James, B.D. Roberts, G. Birkhoff and S.O. Carlsson. We aim to explore the application of orthogonality in normed linear spaces in the best approximation. Hence it has already been proved that Birkhoff orthogonality implies best approximation and best approximation implies Birkhoff orthogonality. Additionally, it has been proved that in the case of ε -orthogonality, ε -best approximation implies ε -orthogonality and vice-versa. In this article we established relation between Pythagorean orthogonality and best approximation as well as isosceles orthogonality and ε -best approximation in normed space.

In this paper, different concept of derivatives with some properties has been introduced. In differential calculus, the partial derivative, directional derivative and total derivative are studied. Their generalization for Banach spaces are the Gateaux differential and Freshet derivative.