Vector Spaces: Basics

This lecture introduces the concept of vector spaces, covering familiar spaces like R2 and RN, as well as less common spaces such as l2(Z) and L2([a, b]). The operational definition of a vector space is discussed, along with formal properties like commutativity and scalar multiplication. Examples in RN illustrate vector addition and scalar multiplication. The lecture also delves into inner products, defining them as a measure of similarity between vectors and discussing properties like orthogonality. The inner product in L2([-1, 1]) is explored, along with norms, distances, and mean square error calculations.