#real-complex-analysis

1 messages · Page 1 of 1 (latest)

cloud flame
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I mean you can look at the metric space as a subset of its completion

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Then the question makes sense, and that's what the setting I interpreted it in

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Well it's a result about representing elements of the dual of a Hilbert space in terms of the inner product

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What you sent is an analog it seems like of the reisz representation theorem

inland drum
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Like Banach space doesn't always have inner product (such as violating parallelogram law), no?

cloud flame
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Sorry I meant hilbert space. The theorem I learnt is if V is a Hilbert space then the map from V to V* given by v -> <v,-> is an isometric isomorphism

glass goblet
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yeah its not that, C_0(X) is not a hilbert space under L^infty norm

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a lot of sources just call this result riesz representation

inland drum
cloud flame
glass goblet
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i prefer to reserve riesz representation for the hilbert space result and call the one for C_0(X) riesz-markov-kakutani

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but many texts will still refer to the C_0(X) result as riesz representation, fortunately it is clear enough from context usually to figure out what theyre saying

inland drum
pine lily
glass goblet
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both of them tend to be called the same thing from my experience which is kind of annoying, but you will know which version to use from context as well

inland drum
glass goblet
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C_c(X) is dense in C_0(X) anyway so they arent that far off

inland drum
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yeah

cloud flame
sage isle
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Let $\Phi:D\subset\mathbb{R}^k\to\mathbb{R}^k$ be 1-1 and $C^1$, where $D$ is a standard $k$-simplex or $k$-cell (for which the closure of $\operatorname{int}(D)$ equals $D$). I also know that open $\Phi(\operatorname{int}(D))$ is dense in compact $\Phi(D)$.\

What I struggle understanding is why $\partial\Phi(D)$ has Lebesgue measure zero as well as the equality $\operatorname{int}(\Phi(D))=\Phi(\operatorname{int}(D))$? Is this clear to you?

nocturne ermineBOT
sage isle
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The subset relation $\Phi(\operatorname{int}(D))\subset \operatorname{int}(\Phi(D))$ is easy, but the other subset relation seems harder.

nocturne ermineBOT
flint igloo
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Um a
I am studying real analysis from abott book and it's summer so I got a lot of time,I am trying to prove theorems on my own...for how long should I try before giving up

south summit
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Positivity is basically the same thing as continuity wrt the LF topology

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Tho not quite

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In the sense that something is continuous wrt LF topology iff it splits into positive and negative parts actually

glass goblet
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strictly speaking im pretty sure the version for positive functionals gets you the general case, but knowing both is useful anyhow

south summit
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Yeah

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It’s a special case of a way more general thing actually

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The Schwartz representation of distributions

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Well tbf you prove that by appealing to this result

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So it’s not actually more general

glass goblet
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i was going to say

south summit
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Anyway

glass goblet
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yeah

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i mean it is still a special case but yeah i dont see how youd get it without riesz-markov-kakutani

south summit
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Yeah yeah

glass goblet
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im under the impression a lot of those theorems actually rely on riesz-markov-kakutani for their proof

south summit
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This rep theorem is hella cool tho

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The schwartz rep thing

glass goblet
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i remember discussing it in an analysis lecture but i dont think we got the proper statement

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i have seen some other representation theorems for distributions in rudin though

tepid bay
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How was this

IMG_3034.jpg
south summit
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I mean all you do is you prove it for compactly supported using riesz rep

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And appeal to a partition of unity

glass goblet
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yeah ok ive seen something similar to that in rudin

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its a partition of unity argument and i think something like that but i forgot the exact details

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i dont remember if he explicitly mentions radon measures but if you have riesz-markov-kakutani i dont think its that huge of a leap

south summit
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Something funny is I think that you should get a similar result for sections of a bundle

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Which is quite funny because of how insanely general it is

glass goblet
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that is beyond my scope so idk

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the representation theorem is nice though since you really arent getting super insane objects when you look at distributions

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everything feels very grounded that way

south summit
south summit
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Or actually idk what the higher version of this statement is

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The 0th order though is that

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A 0th order distribution on C_c(M, E) is a radon measure on the base times a E^* valued measurable form

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Higher order idk tho since it doesn’t fully make sense to take derivatives of distributional sections (or maybe it does eg with a connection, but you don’t get distributions valued in the same bundle)

glass goblet
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interesting

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how much of the standard theory can be generalized to this setting though

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surely you at least want a notion of distributional derivatives

south summit
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I think that you can do it with formal adjoints

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And connections

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You get a derivative of a distribution for every vector field

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Without choosing vector fields you get derivatives from connections but they map into a different bundle

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T^{(0, 1)}M otimes E

tepid bay
magic bridge
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Idk if this is the right spot for this but if $f(n) \in f(\lfloor n/2 \rfloor) + f(\lceil n/2 \rceil) + \Theta(n)$ then we can not necessarily say $f(n)$ is eventually non decreasing right?

nocturne ermineBOT
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Traffic

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magic bridge
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I have more on this but I don't want to repeat it or get noadsed bc that was annoying

charred shell
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Best intermediate level book for complex analysis
For quality problems/interview problems

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And good coverage of theory

turbid hinge
vast ivy
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ive heard bad things of conway's book

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the only four i know of are conway, s&s, ahlfors, and papa rudin, of which ive heard unanimously good things for s&s and ahlfors

turbid hinge
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I guess go with Stein and Shakarchi then

vast ivy
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i wouldnt know lol i havent read any of the four

turbid hinge
vast ivy
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soon 🤞

turbid hinge
vast ivy
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im doing pointset topology and multivar real analysis, then ill do measure theory and complex analysis

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so sometime over the summer

turbid hinge
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I did pointset and measure theory

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but im weak in algebra I need to do more of that

vast ivy
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yeah ive been neglecting it but whatever :p

inland drum
vast ivy
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i see

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ill add conway to my library actually

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all the copies online are fugly though

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like its not clean text

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awesome.

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nvm i found a clean copy

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oh conway has 2 volumes

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i wonder what his 2nd book is about

inland drum
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I heard the complex analysis book by Gamelin is very good tho I haven't read it. Actually, my school uses it for a complex analysis course.

vast ivy
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let me check what the school im attending this fall uses

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ahlfors

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well at least in 2008 they did

inland drum
vast ivy
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i see

inland drum
vast ivy
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whats that

inland drum
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ahlfors won a fields medal

vast ivy
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oh cool

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i didnt know what a field medal was i had to google it

inland drum
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It's like a nobel prize for math.

vast ivy
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i see

inland drum
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the most prestigious awards in math

vast ivy
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oh nice

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seems ahlfors is Cracked

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i mean i presume all of these authors are 😂

inland drum
desert pulsar
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harmonic analysis goat and advised tao opencry