(untitled) (Journal of Advances and Scholarly Researches in Allied Education, 2018)
Vector spaces are the subject of linear algebra and their dimension is well-characterized, approximately defining the number of independent spatial directions. Naturally, in mathematical analysis, infinite-dimensional vector spaces exist as function spaces, which have functions. In general, these vector spaces are endowed with several additional structures, including a topology that permits exploring problems of proximity and continuity. These are more widely used topologies described by a standard or an internal product (with a notion of distance between two vectors). This is especially the case for Banach and Hilbert spaces, which in mathematical analysis are important. Vector spaces in mathematics, science and engineering are increasingly being used. They are the best linear-algebraic definition for systems of linear equations. They provide a basis for expansion of Fourier, used in the compression of images, and provide an environment for partial differential equations for solution techniques.