Elementary Hilbert Space Theory

A Hilbert Space is an inner product space that is complete. Let H be a Hilbert space and for f, g ∈ H, let 〈f, g〉 be the inner product and ‖f‖ = √ 〈f, f〉. (1.1) We will consider Hilbert spaces over C. Most arguments go through for Hilbert spaces over R, and the arguments are simpler. We assume that the reader is familiar with some of the basic facts about the inner product. Let us prove a few things pertaining to limits that may not be so familiar. The inner product and the norm are continuous. That is to say, if un → u, we have: lim n→∞ ‖un‖ = ‖u‖ , 〈un, v〉 = 〈u, v〉 , v ∈ H (1.2)