Jesus, I love you guys. I highly suggest the book by soviet mathematicians on quantum mechanics. The soviets were far more, this is solved let's move on. And worked entirely on integral transforms/actions on hilbert spaces. Theory of Linear operators in Hilbert Space by N.I. Akhiezer & I.M. Glazman.
Here, you'll see matrices as integral transforms; but I really think this will help you guys visualize these results as continuous actions. I know I'm being a broken record here. But all the terms and Matrix reductions you guys are doing, they can entirely be done as integral transforms. And this book does an amazing job of walking you through everything. I wish I could solve all your guys' problem; but primarily, I can identify you guys are actually missing a whole staple in your diet. And I can answer some of your questions, but not all of them.
You mention Normal, but you do not mention the Hilbert space--and that's a red flag to me. We have to remember the "space" the infinite vector exists in. Where the infinite vector space, this infinite vector lives in, has an inner product attached. And from there, we are talking about a Matrix Acting On A Vector Space, which is then Normal/diagonalizable. I agree with everything you guys are doing. But you seem to be missing some key points.
BUT FOR FUCKS SAKES! NOT EVERYTHING IS EUCLIDEAN!!!!!!!!!!!!!!!!!!!!!!!!!!!
Sometimes Infinite vector spaces, that Infinite Matrices act on, don't look Euclidean. And you guys are really missing some things.
Here, you'll see matrices as integral transforms; but I really think this will help you guys visualize these results as continuous actions. I know I'm being a broken record here. But all the terms and Matrix reductions you guys are doing, they can entirely be done as integral transforms. And this book does an amazing job of walking you through everything. I wish I could solve all your guys' problem; but primarily, I can identify you guys are actually missing a whole staple in your diet. And I can answer some of your questions, but not all of them.
You mention Normal, but you do not mention the Hilbert space--and that's a red flag to me. We have to remember the "space" the infinite vector exists in. Where the infinite vector space, this infinite vector lives in, has an inner product attached. And from there, we are talking about a Matrix Acting On A Vector Space, which is then Normal/diagonalizable. I agree with everything you guys are doing. But you seem to be missing some key points.
BUT FOR FUCKS SAKES! NOT EVERYTHING IS EUCLIDEAN!!!!!!!!!!!!!!!!!!!!!!!!!!!
Sometimes Infinite vector spaces, that Infinite Matrices act on, don't look Euclidean. And you guys are really missing some things.