Normed Spaces

If we can manage to define on a set some operations associated with linear vector spaces, we cause this set to acquire a very advantageous algebraic structure. If we can, in addition, endow this set with a topology, we become capable of exploiting all possibilities offered by this structure. The existence of both algebraic and topological machinery on the same set creates a very enriched structure which reveals many interesting properties of such sets. Nonetheless, to render these two structures usefully compatible we have to require that functions representing algebraic operations such as scalar multiplication, addition of vectors be continuous with respect to the topology of the set. Let us recall that we have called such sets in Sec. 4.6 topological vector spaces. In this chapter, we shall consider a particular topological vector space in which the topology on a linear space is generated by a scalar-valued function called a norm which assigns to each vector of the space a non-negative real number. We impose some restrictions on the properties of this function so that it can actually be interpreted as measuring the length of a vector. Furthermore, it becomes possible to define a natural metric on the vector space induced by the norm. Thus, each normed vector space is also a metric space by definition. Therefore, all of Chapter V becomes applicable without exception to normed vector spaces. However, on account of their enriched structure, we naturally expect that normed vector spaces possess some novel properties which are not shared by general metric spaces. Since normed linear spaces are naturally metric spaces, it is reasonable to talk about their completeness. Normed vector spaces which are complete with respect to their natural metric are called Banach spaces. This term was coined by Fréchet. Normed vector spaces play a pivotal part in functional analysis.