Function Spaces – and How They Relate

We’re not interested in a function here or a function there. We’re interested in the collection of all functions to R or to C. Why are we interested in these? Functions describe lots of things. For example, a function on a “physical body” could be used to describe the temperature at every point on the body. Functions to R or C are scalar-valued: they have their image in a field. We are often interested in vector-valued functions, for instance, vector fields on open sets in Euclidean space. However, the theory of vector-valued functions, in the finite-dimensional case, isn’t very different from that of scalar-valued functions, because, after all, the vector-valued function can be described by its scalar components. This raises a number of interesting points that we shall explore as we proceed. Another important point is that for most practical purposes, theorems which work for functions to R also work for functions to C. The main differences are: (1) For functions to R we use the absolute value to define the norm, and for functions to C we use the modulus of a complex number to define the norm. (2) The inner product we take for real vector spaces is a symmetric bilinear positive-definite form, whereas for complex vector spaces, we use a Hermitian inner product.