So I’m having trouble learning about functional analysis, because I feel like a lot of the theorems are sort of extending results from finite dimensions to infinite dimensions, but I don’t have good enough intuition for the richness of infinite dimensional spaces to really appreciate the importance of this.
So I think coming up with a comparison list of theorems which hold in finite dimensions vs. infinite dimensions would be helpful.
Theorems that don’t generalize to infinite dimensions:
-Infinite dimensional vector spaces may not be algebraically reflexive.
-Infinite dimensional vector spaces may not be reflexive under orthogonal complement.
-Infinite dimensional linear operators may be injective and non-surjective, or surjective and non-injective, unlike the finite dimensional case (rank-nullity theorem).
-Weak convergence may not imply strong convergence in non-Hilbert infinite dimensional spaces.
-Orthogonal projection is a unique minimizer.
-I have a vague sense from stackexchange that we can diagonalize the Fourier transform, but I’m not really sure how to think about that.
Do you guys have other big examples you keep in mind for what does/doesn’t generalize? I’m also generally unclear on what implications having an inner product has for Hilbert spaces and how that changes them relative to general Banach spaces.