Regularity of Any Problem

NOTICE

• DO NOT TRUST THE FOLLOWING

Existence and Fixed Points

• We use Hilbert space daily and approximate it from finite dimension vector calculus
• Once our PDE problem is translated into maximization of infinite dimensional vector-valued function, logically the existence of solution is assured (even if it is not computable)
• Two fundamental problems still remain: (1) Is any thinkable PDE problem could be (with adequate topology which form $\sigma$-algebra) turned into finding fixed points in Hilbert space? (2) Can we always find, or does some transcendent principle assure adequate topology to settle arbitrary problem regularized?
• At least two Clay millennium problems are related to above problems: existence and smoothness of Yang-Mills and Navier-Stokes
• In other words, some transcendental version of fixed point theorem, which assures rigorous conservation of energy, is what we seek
• ZFC set theory might be too weak and insufficient to derive such principle