functional analysis and real analysis

[leaden shuttle]

Need help with it

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[serene olive]

what about it specifically lol

[leaden shuttle]

I have on banach spaces and so on its multiple topics

[serene olive]

can u just post the question?

it might be easier that way

[leaden shuttle]

Ok

[serene olive]

also im a bit new to this too so cant guarantee lol

[near rapids]

still need the question 1st before anything else lol

[leaden shuttle]

Banach spaces what is it ...
And it's properties or applications related to it

[serene olive]

thats a very general statement

can you be a bit more specific

[leaden shuttle]

My problem in on the entire banach spaces topic

[near rapids]

A banach space is a complete normed vector space

[serene olive]

so what is it

[leaden shuttle]

Carry on

[near rapids]

that's it

that's all a Banach space is

[serene olive]

you need to ask specific questions or we cant help

[leaden shuttle]

What is meant by complete normed vector space

[near rapids]

well I assume you know what a vector space is

and what a norm is

[serene olive]

do you know what each of those words mean

[near rapids]

^

[leaden shuttle]

Yes

[near rapids]

complete means cauchy sequences converge

[leaden shuttle]

Ok

[serene olive]

its just definitions so you can open p much any textbook

[near rapids]

the definition is literally know the 4 words and put them together

naively, a Banach space is a space where convergence is akin to R

[leaden shuttle]

Ok hold in I'll have some more questions

[near rapids]

(since R is complete wrt |.| as you learn early on in analysis)

[leaden shuttle]

Can u help with all these questions on how to prove them step by step

@near rapids

[near rapids]

no

[leaden shuttle]

?

Why

[serene olive]

[near rapids]

my functional course wasn't good, and if you're questioning a very basic definition I somewhat doubt you'll be capable of doing questions

basic as in fundamental*

[leaden shuttle]

I think I can clear then when we go through the way

Cz the definition is not that hard

I have problems in how to start the process

[serene olive]

you need to sit down with a textbook and learn it urself lol

[leaden shuttle]

And why do I have to assume certain types

[leaden shuttle]

I don't want to learn it rather I want to know

[near rapids]

how do you expect to know if you dont learn

[leaden shuttle]

I mean I wanna learn

[near rapids]

part of math is just trying stuff and seeing what comes to fruition

[leaden shuttle]

But not like learning a poem

Or such

[near rapids]

lot of times it boils down to writing out the definitions and piecing it together

[leaden shuttle]

What I wanna say is in such questions how do I understand what I want to do

And what should be the first step

[near rapids]

.

1st step

write down the definitions needed

[leaden shuttle]

Cz books simply give the proof

But now why the first step was taken

[near rapids]

you.. read the proof

and see

what they did first

[leaden shuttle]

Isn't there a way to approach it without know the proof and constructing the proof in spot

Cz the question given are not present in any book

[near rapids]

lol what

how do you expect to prove something without making the proof?

[leaden shuttle]

It more like have to make it on the spot with the info we have

[near rapids]

yeah

welcome to math, you're doing real math

[leaden shuttle]

That's what I said we make the proof not look what they proved

[near rapids]

you look at proofs already done to learn the arguments typically needed

[leaden shuttle]

Yes

That's done

But I have problem in this problem set i provided

Can u help with question 1d

[near rapids]

Just follows from the inequality of p-norms

[leaden shuttle]

Ok

Which inequality

Norm x less equal to 1

[near rapids]

$1\leq p\leq q<\infty\implies \norm{x}_q\leq\norm{x}_p$

[gilded silo]

Omegabet_

[leaden shuttle]

Ok

Now

[near rapids]

Now... attempt it

[leaden shuttle]

How do I show its bounded

[near rapids]

what does it mean to be bounded?

again, unpack definitions

or just write out the definition of norms and see what you can do

[leaden shuttle]

There exist some M>0 s.t
Lp norm x =< M supnorm x

[near rapids]

$\norm{x}_1\leq 1\implies \int_0^1\abs{x(t)}\dd{t}\leq 1=\int_0^1\dd{t}$, so $\abs{x(t)}\leq 1$

[gilded silo]

Omegabet_

[near rapids]

from writing out definitions

[leaden shuttle]

Ok

So the area is less than 1

[near rapids]

now, crazy idea, write out what the infinity norm of x is by definition

and note that, wow, writing definitions as scratchwork, is kinda useful

[leaden shuttle]

Ok

Sup norm x = sup {|x|}

[near rapids]

$\norm{x}\infty=\max{0\leq t\leq 1}\abs{x(t)}\leq\max_{0\leq t\leq 1}1$

[gilded silo]

Omegabet_

[leaden shuttle]

Ok

Now

[near rapids]

so the set is bounded.

since I assume you know how to argue the max of 1 is 1

[leaden shuttle]

Hmm but here is a question

[near rapids]

waiting

[leaden shuttle]

Geometry if we look

Sup of x is the height

But the area reduces

The lp norm

[near rapids]

we're not in $\ell_p$

[gilded silo]

Omegabet_

[near rapids]

we're in function space

[leaden shuttle]

Yes I know

What I mean is lp norm

Or p norm

[near rapids]

call it what it is, yes

ell_p is something else

[leaden shuttle]

Hmm

[near rapids]

again, the proof was literally write out definitions

and then piece it together

that's kinda the theme in a lot of analysis proofs

[leaden shuttle]

Let me provide u what I did

[near rapids]

yeah idk what that;s suppose to do

you look like you took a subset of the set and showed it was bounded

not the entire set

[leaden shuttle]

What I found in my way is unbounded

[near rapids]

ok and I dont follow your proof, so that's the main issue

[leaden shuttle]

Cz the subset is defined on p norm

And the parent space is defined on sup norm

[serene olive]

ur right its unbounded

[leaden shuttle]

What I mean is that the area

[serene olive]

thats the right intuition. you can have small area but have the function be large

[leaden shuttle]

Redused as the base is shortened

[serene olive]

because the spike is big

[leaden shuttle]

Yed

Yes

[serene olive]

epic

[leaden shuttle]

Btw can I reply on chat gpt or bard

Rely

[serene olive]

no

dont use chat gpt for maths

[leaden shuttle]

Sometime they provide wrong ans right

[serene olive]

yes

because they dont always interpret the question right

[leaden shuttle]

Ok

Ok let me first rebrush some defination

And then I'll and u for help with some questions

Or else it problametic for me

Have you taken a course on topology I would expect that to typically be prereq for functional analysis? And banach spaces should have been covered then. Could be wrong tho but I do find it a bit odd

[serene olive]

for me, topology does not use the linear structure that you find for banach spaces

[leaden shuttle]

Yes i had topology in my previous course

And I have a back on it

[leaden shuttle]

True true but you often cover it as a side topic no?

I understand but I'm not good at writing in the proper language
And the professor deducts the marks even if the idea is right

[serene olive]

not for me actually

we have functional analysis split up into 2 courses: linear analysis (banach and hilbert space, spectral theory) and analysis of functions (TVS, distribution theory etc.)

[leaden shuttle]

Ok maybe different books I suppose mine had some operator theory after point set top

Ok ignore my messages then op you'll just need some practice revising the definitions and all ig

[serene olive]

i think books tend to cover a lot more material than say a 24 hr lecture course lol

[leaden shuttle]

That's probably linear operator T on functional analysis

Analysis is very disgusting cz I don't seem to write it well on things I understood

[serene olive]

if you write enough proofs you will find it natural

humans aren't designed to do analysis so you should find it hard to start

[leaden shuttle]

Yes

Thanks for understanding

♥️