Mutually Singular implicit functions

Abstract A continuously differentiable (or smooth or in 𝐶 ′ (𝑋1 )) function 𝜋1from an open set 𝛤 which is a subset of a linear space X1 of dimension 2n into a linear space X2 of dimension n has an inverse onto a subspace upto equal dimension. If (s, t) is in 𝛤 where 𝜋1 (𝑠,𝑡) = 𝟎, 𝜋1 ′ (𝑠,𝑡) = 𝑀invertible at (s, t), then 𝜋1is invertible in the neighbourhood of (s, t) 𝜖𝛤. Upto bijection, the invertibility of a multi variate function 𝜋1is possible and for this 𝜋1 from X1 is mapped onto X2 so as to get invertible for the 1st n components of the 2n – tuple. In the present discussion, we focus on two multi variate functions 𝜋1 and 𝜋2 or linear mappings from X1 into X2 such that both are smooth in two open sets 𝛤1 and 𝛤2 subsets of 𝑋1 taking the 1st n – components of 2n – tuples in X1 onto 𝑋2 while 𝜋2 is from the open set 𝛤2 of 𝑋1 taking the 2 nd set of n – components of 2n – tuples in X1 such that 𝜋1 ′ is invertible at (𝑠1,𝑡1 )𝜖𝛤1 and 𝜋2′ is invertible at (𝑠2,𝑡2 )𝜖𝛤2 . The sum of derivative of the implicit functions is a continuously differentiable function gives 𝜌1′(𝑡) + 𝜌2′(𝑠) = - {𝑀1𝜉 −1𝑀1ή + 𝑀2ή −1𝑀2𝜉 } ...... 0.1 with 𝜌1 and 𝜌2 are the implicit functions, 𝑀1 = 𝑀1𝜉 + 𝑀1ή and 𝑀2 = 𝑀2𝜉 + 𝑀2ή in which 𝑀1𝜉 and 𝑀2ή are the invertible parts while 𝑀1ή and 𝑀2𝜉 are singular parts. Further, the null space of 𝜋1 is the range space of 𝜋2 and vice versa leading to 𝜋1 and 𝜋2 are mutually singular, denoted by 𝜋1┴𝜋2 ......0.2 Key words: linear space, multi variate function or a vector valued function, nonsingular mapping, mutually singular mapping, invertible mapping, null space 𝜗(𝜋), range space 𝛾(𝜋), dimension of a linear space, span, rank and nullity of a linear transformation, contraction, upto invertibility, subset, open mapping.