Today I shall speak on a topic that I feel is important.
All of us have encountered the “” norm in Functional Analysis. In , for . In the dual space , . What is the utility of this “sup” norm? Why can’t our norm be based on “inf” or the infimum? Or even ?
First we’ll talk about . Let us take a straight line on the plane, and a sequence of continuous functions converging to it. How do we know they’re converging? Through the norm. Had the norm been , convergence would only have to be shown at one point. For example, according to this norm, the sequence converges to . This does not appeal to our aesthetic sense, as we’d want the shapes of the graphs to gradually start resembling .
Now let’s talk about . If we had the norm, then we might not have been able to take every point and show that the sequence is a cauchy sequence. So what if we cannot take every and say is a cauchy sequence? This would crush the proof. How? Because then would not resemble the terms of the cauchy sequence at all points in , and hence we wouldn’t be able to comment on whether is linear for all or not. Considering that contains only bounded linear operators, the limit of the cauchy sequence not being a part of would prove that is not complete. Hence, in order for us to be able to prove that is linear and that is complete, we need to use the norm.