Foreword

What is functional analysis? A satisfactory answer requires going back to where it all started.

"All are present; the meeting convenes," intoned Fredholm. Intent were the gathered faces, their thoughts fixed on their students. "What do we know of their weaknesses?"

Hilbert leaned back, torch's light flickering across his features. "Lots of dimensions, especially when they need to find the Hessian. What if… what if we made them deal with infinitely many dimensions?"...

It was Banach who finally spoke. "David, they already know about the vector space for the polynomials".

Hilbert smirked. "Who said anything about countably infinite?". More silence, then glances, then grins.

It was Riesz's voice which next broke the silence. "And we can make them do analysis in that space. And linear algebra, but not the easy parts. Of course, they'll need to also deal with complex numbers. Sprinkle a little topology and abstract algebra on top, because... they deserve –"

"Frigyes, some of them might actually be able to do that. We need more." After a pause, Fredholm continued: "We'll tell them that they only need to know basic calculus."

A Friendly Approach to Functional Analysis

I didn't actually find the book overly hard (it took me seven days to complete, which is how long it took for my first book, Naïve Set Theory), although there were some parts I skipped due to unclear exposition. it's actually one of my favorite books I've read in a while – it's for sure my favorite since the last one. That said, I'm very glad I didn't attempt this early in my book-reading journey.

My brain won't stop line to me

Some part of me insisted that the left-shift mapping

$(x_{1}, x_{2}, \dots) \mapsto (x_{2}, x_{3}, \dots) : ℓ^{\infty} \to ℓ^{\infty}$

is "non-linear" because it incinerates $x_{1}$ ! But wait, brain, this totally is linear, and it's also continuous with respect to the ambient supremum norm!

Formally, a map $T$ is linear when $T (α x + β y) = α T (x) + β T (y)$ .

Informally, linearity is about being able to split a problem into small parts which can be solved individually. It doesn't have to "look like a line", or something. In fact, lines^[1] $y = m x$ are linear because putting in $Δ x$ more $x$ gets you $m \cdot Δ x$ more $y$ !

Linearity and continuity

Two things surprised me.

First, a(n infinite-dimensional) linear function can be discontinuous. (?!)

Second, a linear function $T$ is continuous if and only if it is bounded; that is, there is an $M > 0$ such that

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