What's a Banach Space? How do I begin understanding Banach Spaces?
>>7670953
Yeah that was my old thread but not a single serious response only le memeshit
https://www.youtube.com/watch?v=n12bfWTw9Hk
A Banach space is a normed vector space V (ask if you don't know what it is) that is complete (ie. such that every Cauchy sequence in V is convergent).
Banach spaces are interesting because we have a number of very powerful results allowing us to prove the continuity of linear maps without much effort (Banach-Steinhaus, open mapping theorem, Hahn-Banach)
>>7670925
I don't know, but the creator of dwarffortress wrote a thesis about flat chains in them. thats something
http://arxiv.org/abs/math/0411309
>>7671070
I know what a vector space is, but what's a normed vector space?
>>7671087
Its one that's normal :^)
>>7671087
you have a norm, i.e. some kind of distance.
>>7671091
Is Banach Spaces a Linear Algebra topic?
>>7671101
Everything is a linear algebra topic.
>>7671101
Banach Spaces are Functional Analysis.
>>7671101
Finite dimensional Banach spaces are not very interesting, they are basically R^k, so they are indeed studied in Linear Algebra.
>>7671115
So what are infinite dimensional Banach spaces?
>A Banach space (over or ) is a complete normed vector space. If X is a vector space with two different norms, · and · , then the norms are said. to be equivalent if there are positive constants C1 and C2 such that. C1 x ≤ x ≤ C2 x. for all x ∈ X
here you go OP
>>7671120
Mostly function spaces, i.e. they are studied in functional analysis.
Make my foot converge to you ass in the Cauchy sense! xD So random!! /r/math
>>7671145
It's because of the meme vid and now I'm really curious but I'm not that smart so there's that.
>>7671120
Functional analysis, most likely.
The important property of finite dimensional normed linear space is that they are locally compact (it's actually equivalent to being finite dimensional) and, in particular, are all complete.
That's why we don't really talk about Banach spaces in the finite dimensional case (it's automatically satisfied).
Another property of finite dimensional normed linear spaces is that all norms are equivalent (ie. whatever norm you choose, the notion of convergence is preserved), therefore there is only essentially one topology of normed space per dimension.
In the infinite dimensional case, you can put several fundamentally different norms on a vector space and get different informations.
The reason Banach spaces are studied in functional analysis is that most interesting and natural examples of Banach spaces are function spaces (Lp spaces, test functions, Sobolev spaces).
However, the study of topological vector spaces really is linear algebra (for example, spectral theory is essentially the continuation of the study of diagonalization in the finite dimensional case). The only difference is that in the infinite dimensional case, we have to restrict ourselves to continuous linear maps in order to say interesting things (in the finite dimensional case, all linear maps are automatically continuous), hence the importance of topology.
>>7671161
You just need to know Linear Algebra and Real Analysis to study Functional Analysis.
After learning this best start with the book by Rudin before moving to harder Functional Analysis books like the one by Yosida and then Zeidler's book series for Nonlinear Functional Analysis.
>>7671175
I know Linear Algebra and only some basic Real Analysis (sequences, compactness, really basic topology with neighborhoods and stuff), is that enough knowledge to start reading that book in the pic?
>>7671181
No you need to study every waking moment for 30 years to even begin to understand this book.
>>7671196
My sides
>>7671181
Probably not, you don't have enough mathematical maturity. Pick up a book like "introduction to functional analysis" or "basic functional analysis".
Basic functional analysis is usually covered in an actual Real Analysis course(lebesgue measure).
>>7671203
>an actual Real Analysis course
I'm getting that next year but I want to know everything NOW
>>7670925
Introductory Functional Analysis with Applications by Erwin Kreyzig
>>7671216
Functional analysis is pretty meh to be honest family. Unless you're into physics I guess. I would concentrate my studies on Algebra and Topology. Analysis is going out of fashion fast.
A Cauchy series is a series of numbers (in [math]\mathbb{R}[/math], or [math]\mathbb{R}^n[/math] or whatever space you can measure lenghts in) where at some point the difference between all following points is arbitrarily small, meaning given a small value [math]\epsilon[/math], there's a number so that all n-th elements coming after the N-th element will have a distance smaller or equal to [math]/epsilon[/math]
A (cauchy) series converges when there's one point in that space where the series comes arbitrarily close to, meaning that within a "sphere" of arbitrarily small radius around that point almost all elements are cointained. To make this clear think about a series in an x-y coordinate system that spirals closer and closer into the (0,0) point. I can draw a circle at any length, and except for a few finite-number elements all the other points will be within that circle. And if I make the circle smaller it's still true, it's just that the number of finitely many points outside has increased a bit.
So this is what convergence means and a space is called a Banach space, if all cauchy series converge on it. This sounds a bit technical and/or obvious, but there are spaces were Cauchy series don't always converge.
Banach spaces are an important too in mathematics. For example every Hilbert space (which can describe the states of wave functions in quantum mechanics) is a Banach space.
>>7671264
Intredasting. I know [math]\mathbb{Q}^{n}[/math] isn't a Banach space because we have Cauchy sequences that don't converge to a rational number. Is that correct?
>>7671268
Intuitivey i'd say yes, but it's been a while since by Calc I course so I can't be certain.
The reasoning is that because the irrationals are dense in [math]\mathbb{Q}[/math], you could define a series of rational numbers that approximates an irrational number. For example:
3
3.1
3.14
3.141
3.1415...
looks like a cauchy series (proof left to the reader kekeke) that would approximate [math]\pi[/math] to arbitrarily small length (where length just means the standard difference between two numbers). It seems to converge to [math]\pi[/math], but we must remember that [math]\pi[/math] is irrational, so it's NOT an element of [math]\mathbb{Q}[/math], so the point of convergence isn't in [math]\mathbb{Q}[\math] and the series doesn't converge in [math]\mathbb{Q}[\math].
>>7671231
>Analysis is going out of fashion fast.
Why's that?
I did Hilbert spaces in 2nd year physics, not to bad honestly
>>7671296
not really sure. one main point is obviously algebraic objects have a nice combinatorial structure.
>>7671231
Just because /sci/ only memes about arithmetic geometry doesn't mean that analysis is "going out of fashion".
There are still many very much alive fields of math with strong links to analysis (PDE, of course, ergodic theory, number theory and a number of areas of algebraic geometry)
