Bounded linear operators on ∞-dimensional vector spaces
A matrix is a box filled with numbers
with the context understood to be that they will be multiplying something in an inner-product sense.
i.e. "matrix on the left" is read with an arrow → going right across rows. "matrix on right" is read with an arrow ↓ going down columns.
If you read about spectral theorems or bounded linear operators or even just abstract vector spaces you might come across, as I did, mention of "infinite-dimensional spaces". What could that even mean? How do the dimensions fit together? How can I picture an infinite-dimensional thing?
I recently learned the answer and it's not nearly as hard as I thought; I'll share my new perspective with you.
Normally we talk about an entry a_{i,j} in the matrix. It's indexed by {row,column} where i,j ∈ {1,…,N}.
The "infinite-dimensional vector space" idea uses the same a_{i,j} but i,j ∈ [0,1], the continuous line segment which bijects to [1,N] (another continuous line segment--just shift back by one and divide to biject it)
So the matrix entries function the same way, they're just now to be thought of as "continuous rows"
…and the eigenvectors (discrete entries) become eigenfunctions (attaining values from continuous scale)
--if you have the mental machinery to envisage a probability distribution--even better a 2-D joint distribution--then you have what's required to "picture" this thing.
If you picture each of the matrix "blocks" as corresponding to a light/darkness value to represent the quantity inside
then the "infinite-dimensional" linear operator would just be "more subsquares in the grid". If you want to allow complex values then Elias Wegert's pictures (using colour as a "circular" value (complex argument) rather than brightness as a "straight" value) then his pullbacks on a complex square Rect(Z)→arg(f) (I used 1000×1000 resolution) look fairly continuous--like an infinite-dimensional linear operator taking complex values a_{i,j}∈ℂ, i,j ∈ [0,1]
That's the formal aspects taken care of. What kinds of things might an infinite-dimensional space be needed to represent? Here are some ideas:
handwriting -- can we capture all possible forms of the letter "a" with a finite number of knobs?
temperature on a plate -- if you don't want to think on an atomic level (or use a mole×mole dimensional matrix to talk about something) you could imagine a 2-D plate being a continuous box attaining different temperatures at different continuous points