Parseval or Parsifal?

Which takes longer, proving Pythagoras’ theorem in infinite dimensions (Parseval) or in finite? The question sounds a little silly, but really it is a challenge that keeps coming up. The problem is to decide which out of several perspectives is somehow more basic, and the choice of axioms will then determine which theory is more complicated. There are fewer assumptions made for Parseval’s identity that x 2 = ∑ α ∈ A x x α 2 when ⋅ ⋅ is an inner product and the span of the x α is dense in the space (so, when the space is also complete, that is Hilbert, the span is the whole space). On the other hand, you might feel that finite dimensional linear algebra is somehow more simple because it can be visualised.

The question has some substance because of the way we perceive what we are proving, and how it relates to our intuition. The simpler things are not necessarily the things we come across first, nor does the difficulty in discovering something always relate to how long it takes to be discovered. I think most mathematicians vaguely feel like there is some quest to build higher knowledge, putting more proofs on the top, always building up to further generalisations and then filling in the detail of the new structures below (like an inverted pyramid). On the other hand, simple things like this challenge that metaphor. Perhaps we are climbing up a tower already there, while others are digging away at the foundations and finding they keep on going down. Alternatively, my tea-break yesterday reading the introduction to Kuhn’s book was may have just given me too much melodrama.