Why is one in functional analysis only investigating complete spaces (like Banach or Hilbert spaces)? I heard someone saying that analysts in general like to work with limits, which makes sense. But what would be more satisfying is to see a concrete situation in the theory where completeness is essential. Also, why is it essential in the applications to differential equations?
And anyways: Is there not much to say about non-complete normed spaces? Have people investigated them?