Hi.
Why do we need proof of the Cauchy-Schwarz Inequality? Why couldn't the statement be accepted as a [word for statements which didn't require proof when they were first stated such as shortest distance between two points is a straight line] like most other obvious things?
>>7952985
>word for statements which didn't require proof when they were first stated such as shortest distance between two points is a straight line
Surely there's a word for that thought
>>7952985
One of the requirements for a norm is that the Cauchy-Schwartz inequality holds. For something to be a norm you must show that the inequality holds.
If you're talking about absolutes in Analysis, then it must be proven because you can't just assume the absolute value is a norm. Once you just start assuming things, you start fucking up your math.
Shortest distance between two points is a straight line is something that needs to be proven. You can come up with geometry where that is not true. That point highlights my point that things need to be proven, no matter how "obvious" it might seem to you.
>>7953160
>norm
You're confused about what this word means
If we dont then for one, we havent the foggiest idea of whether or not the equality will hold which is bad in and of itself. But another consequence which would be almost worse, is that every proof using the Cauchy-Schwarz inequality will be met with the infuriating question "What if it doesnt hold for [Insert arbitrary case here]?".
>>7952996
perhaps an AXIOM
>>7952985
axiom
>>7953167
He means a metric but it's clear enough.
>>7952985
>shortest distance between two points is a straight line
That one can be actually proved.
>>7953167
I'm not that guy, but I guess the authors of the metric spaces books I use to study are also confused about the meaning of that word.
>>7955311
Literally the Cauchy schwarz inequality - the topic of this thread...
>>7952985
"Most other things" aren't just accepted.
My analysis class only assumed 4 set theory axioms, not even the full ZFC
Also, nothing is "obvious" in math. It exists independently of your experience and doesn't give a flying fuck what you think is obvious.
>>7955391
>It exists independently of your experience
found the undergraduate
>>7955422
I'm studying measure theory rn for my graduate classes... Sorry friend.
To suggest that a logical system depends on real world experiences is bullshit.
Tell me, what are you doing in graduate school?
