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## Main Question or Discussion Point

I'm trying to really solidify my maths knowledge so that I'm completely comfortable understanding why and how certain branches of mathematics are introduced in physics and inevitably that leads me to a study of proofs. I usually skip proofs as I found them annoying, unintuitive and just seemingly tedious guess work.

I started a course on analysis at uni for my physics degree but switched option because I hated it so much.

For example The Schwarz Inequality proof in a book I'm using:

For me however, I immediately get stuck on where [itex]|c\rangle[/itex] came from.

I tried to prove this on my own multiple times and probably spent about 6 hours total on it on and off and end up messing about writing [itex]|a\rangle[/itex] and [itex]|b\rangle[/itex] in terms of orthornormal vectors before feeling like an idiot and ultimately getting nowhere and just giving up.

If left to work on this problem for a year i can't even imagine what on earth would cause me to think of introducing a third vector [itex]|c\rangle[/itex] into the mix.

Then, not only that, but assigning it the very specific value:

[itex]|c\rangle = |b\rangle - (\langle a|b\rangle / \langle a|a\rangle)|a\rangle[/itex]

I'd have a better chance of winning the lotto than happening to come up with this.

Clearly I'm lacking something fundamental here, I'm guessing the choice of the vector c might seem obvious to someone good at this, but in my mind the great majority of non trivial proofs proceed in the following way:

-> 1) Algebra that I can follow clearly

-> 2) Insert step that seems to have been conjured out of thin air (ie lets introduce the vector [itex]|c\rangle = |b\rangle - (\langle a|b\rangle / \langle a|a\rangle)|a\rangle[/itex])

-> 3) Algebra that I can follow clearly

-> 4) QED.

I then sit there for ages trying to justify how I would come up with step 2 and, i sort of can see why, but it is just 99.9% hindsight because once it comes time for me to do practise proofs on my own, I still don't even know where to start, and when i look for the proofs online, the above scenario plays out again. This repeats until I just eventually get fed up!

This isn't my first foray into trying to learn these things, but I never get further than this and end up just deciding to just accept proofs and move on because its just a massive waste of my time and has seemingly little if any benefit.

I started a course on analysis at uni for my physics degree but switched option because I hated it so much.

For example The Schwarz Inequality proof in a book I'm using:

From which point it is clear the inequality can be proven.For any pair of vectors [itex]|a\rangle[/itex],[itex]|b\rangle[/itex] in an inner product space V, the schwarz inequality holds:

[itex]\langle a|a\rangle \langle b|b\rangle \geq |\langle a|b\rangle |^2[/itex]. Equality holds when [itex]|a\rangle[/itex] is proportional to [itex]|b\rangle[/itex]

The proof then starts:

Let [itex]|c\rangle = |b\rangle - (\langle a|b\rangle / \langle a|a\rangle)|a\rangle[/itex]

Therefore [itex]|b\rangle = (\langle a|b\rangle / \langle a|a\rangle)|a\rangle + |c\rangle[/itex]

Then take inner product of [itex]|b\rangle[/itex] with itself:

[itex]\langle b|b\rangle = \left|\dfrac{\langle a|b\rangle}{\langle a|a\rangle}\right|^2 \langle a|a\rangle + \langle c|c\rangle[/itex]

For me however, I immediately get stuck on where [itex]|c\rangle[/itex] came from.

I tried to prove this on my own multiple times and probably spent about 6 hours total on it on and off and end up messing about writing [itex]|a\rangle[/itex] and [itex]|b\rangle[/itex] in terms of orthornormal vectors before feeling like an idiot and ultimately getting nowhere and just giving up.

If left to work on this problem for a year i can't even imagine what on earth would cause me to think of introducing a third vector [itex]|c\rangle[/itex] into the mix.

Then, not only that, but assigning it the very specific value:

[itex]|c\rangle = |b\rangle - (\langle a|b\rangle / \langle a|a\rangle)|a\rangle[/itex]

I'd have a better chance of winning the lotto than happening to come up with this.

Clearly I'm lacking something fundamental here, I'm guessing the choice of the vector c might seem obvious to someone good at this, but in my mind the great majority of non trivial proofs proceed in the following way:

-> 1) Algebra that I can follow clearly

-> 2) Insert step that seems to have been conjured out of thin air (ie lets introduce the vector [itex]|c\rangle = |b\rangle - (\langle a|b\rangle / \langle a|a\rangle)|a\rangle[/itex])

-> 3) Algebra that I can follow clearly

-> 4) QED.

I then sit there for ages trying to justify how I would come up with step 2 and, i sort of can see why, but it is just 99.9% hindsight because once it comes time for me to do practise proofs on my own, I still don't even know where to start, and when i look for the proofs online, the above scenario plays out again. This repeats until I just eventually get fed up!

This isn't my first foray into trying to learn these things, but I never get further than this and end up just deciding to just accept proofs and move on because its just a massive waste of my time and has seemingly little if any benefit.