What's a Banach Space?
https://www.youtube.com/watch?v=n12bfWTw9Hk
A Banach space is a complete normed space.
For example the real numbers with the absolute value function as norm or the space of continuous functions from the interval [a,b] to R with the supremum norm.
>>7700831
What the fuck is a norm?
>>7700835
A function from a vector space V into the real mumbers such that for all scalars r and all vectors v,w we have
|v| >= 0
|v| = 0 iff v=0
|rv| = |r| |v|
|v + w| <= |v| + |w|
>>7700837
Like the length of a vector?
And complete means that all Cauchy sequences converge right?
What's the use of Banach Spaces?
Not that guy, but I've been meaning to ask this for a while:
Wikipedia says:
"A Banach space is a vector space X over the field R of real numbers, or over the field C of complex numbers..."
Must the ground field be R or C, or will other fields (e.g. Q) do?
>>7700860
OP here, [math]\mathbb{Q}[/math] WON'T do because it isn't complete. We can define a Cauchy sequence in [math]\mathbb{Q}[/math] that's NOT convergent in [math]\mathbb{Q}[/math].
>>7700864
p-adic Banach spaces exist, though.
>>7700845
>Like the length of a vector?
Yes, you define the length of a vector as it's norm.
>And complete means that all Cauchy sequences converge right?
Yes.
>What's the use of Banach Spaces?
In Banach Spaces it's easier to prove things like the existence of solutions of differential equations or optimization problems than in general normed vector spaces.
>>7700874
>Where can I learn more about this?
In Functional Analysis but you need good knowledge of Real Analysis first.
>>7700864
Get out now or I will make my foot converge to your ass in the Cauchy sense.
The fuck? A meme turned into an actual thread?
This is spoopy /sci/...
Also, what previous knowledge is necessary for a real anal course?
>>7701075
not much, mostly set theory and series. You have to know how to do induction proofs and use inequalities (that you proved by induction before).
Most of the course will be about proving a bunch of theorems using tricks, and then you have to do a bunch of problems using those theorems as tools. It's a pretty frustrating course because its gonna seem like the problems are impossible, but you just need to do lots of them and hope to christ that a similar one will be on the exam.
>>7700837
>|v| >= 0 (1)
|v| = 0 iff v=0 (2)
|rv| = |r| |v| (3)
|v + w| <= |v| + |w| (4)
(1) is redundant as it follows from the other three rules: apply (4) with w = -v, then (3) with r = -1, then (2).
>When she licks the pre-Banachspace from your tip
>>7700872
>In Banach Spaces it's easier to prove things like the existence of solutions of differential equations or optimization problems than in general normed vector spaces.
it is easier only because we manufacture the space in which it is easy.
this is why math is fruitless.
>>7701099
Yeah man. Maths doesn't have any applications I agree.
>>7701088
>(1) is redundant as it follows from the other three rules: apply (4) with w = -v, then (3) with r = -1, then (2).
the point is that it is [2] which is generalizable which destroy your theorem
>>7701082
My knowledge right now is about sequences (limsup, liminf, Cauchy sequences), series, set theory, continuity, integral calculus, differentation, and power series. Do you think I'm ready?
Any book recommendations?
>>7701101
you do not know why you do math.
>>7700825
Warwick second year detected
