Introduction to vector spaces generalises all vector like quantities. An arrow tipped representation is not necessary. The vector addition and scalar multiplication amounts to superpositions, like superpositions of waves. A consonant cannot be pronounced without being accompanied with a vowel ; a vector cannot be written without being multiplied with the scalar 1. The comes the approximation of functions by Bessel's series or Fourier series. Vector spaces unify all such special functions which are required to express the solutions of differential equations which cannot be expressed in closed forms.Also Laplace's and Fourier transforms etc.are well treated in vector spaces which transform differential equations into algebraic equations which are easier to solve and their inverse transform becomes solutions of differential equations; just like logarithms transform a problem of multiplication into addition whose anti log gives the product. This introductory book shall summerise the topics in foundation courses in Mathematics taught in Engineering disciplines and graduate classes and help in comprehension rather than making them o0bscure by summerising.
Two volumes are planned planned next; one for treatment of serries or special functions and transforms and another on linear transformations and matrices defined in vector spaces.which reveal the structure of the vector spaces and have vast and varied applications. The edifice of quantum mechanics stands upon this.