Boundedness Puns

I am almost finished my dissertation and I need to prove that a map is continuous, for which I need to use the fact that ||Tx||<= C||x|| (boundedness) implies that T is continuous for a linear operator T.

I wrote the theorem but not the citation, and dont have time to be rifling through books right now, I have so much more work to do! Please let me know if you know where I can find this.

Thanks in advance x

Is it possible for a metric space of cardinality greater than the reals to be bounded?

I was just thinking about this idly, and I don't know enough set theory or analysis to really come to a conclusion. The completion of a metric space can be seen as "filling in the holes," and the completion of a complete space is the same space. So we've completely filled the holes.

All bounded metric spaces can be completed to be the cardinality of the reals, with nothing left to "fill in," so my intuition is that a bounded metric space cannot have cardinality greater than the reals.

Can someone confirm/deny?

Prove: The sequence a(n) = n^2/(3^n) is bounded.

Attempted Direct Proof: Let K=1 and n arbitrary, then

|n^2/(3^n)|= n^2/(3^n) [As n^2/(3^n) >= 0 for all n]

n^2/(3^n) <= n^2 <= ...?

(Definition: A sequence a(n) is bounded if there is some K ∈ ℝ s.t. for all n ∈ ℕ |a(n)| ≤ K.)

Thanks for any hint!

Hello all, I will link the question and some extra info but the only question I'm struggling with the part involving uniform boundedness theorem, showing that for our T_n there is a c_x > 0 etc.

Here is our set of functions : https://gyazo.com/04c395fbdf0b60590656d43e0d2b3224

And here is the question I'm struggling with (2ii, I've done 2i):

https://gyazo.com/38f129f77a7ae0ca64470efb53865fdb

The only example I have using the uniform boundedness theorem shows that a normed space is not a Banach space, but I'm not sure how to apply that to the context of this question. Thanks!

Without using a graph

Hello again, good to be back. I only have two problems this time (I've been working on some more trivial examples over the weekend: continuity, simple limits, convergent and divergent sequences of numbers, etc.), but I hope that they're gotten to.

https://imgur.com/a/mw7w2lQ

2.50 deals with proving the sup-norm of a function isn't infinite iff f is bounded. Seemed simple, just curious if I understood the definitions mainly.

2.62 is where I need help however. I'm still working with uniform convergence and pointwise convergence, and I think I'm understanding how to prove something is uniformly convergent. I'm still a bit unsure on how to disprove it however. I revised this problem a few times, but I think the last image is when I finally caught how to disprove uniform convergence of a sequence of functions.

The images are in order, where the first two deal with 2.50 and the remaining are 2.62. Thank you in advance for your help, I haven't posted in a few days since most of my studying has been reading sections, and reviewing simpler definitions and theorems.

For Riemann sums, we have both left and right Riemann sums, if we consider the interval [a, x], then the left Riemann sum of some function, f(x), with ∆x=(x-a)/N is lim_{N \to \infty}∆x[f(x)+f(x-∆x)+f(x-2∆x)+...+f(a+∆x)], so for the term f(a+∆x), we never actually get to f(a), so we don't need f to be bounded at a, only in (a, x]. I apply the same reasoning to the right Riemann sum lim_{N \to \infty}∆x[f(x+∆x)+f(x-2∆x)+f(x-3∆x)+...+f(a)], so in the right Riemann sum f has to be bounded on [a, x). But, in the limit these should be equivalent, so how can the requirments be different? Also, it wouldnt make sense for the function to need to be bounded on [a, x], since then for example this would imply 1/sqrt(x) integrated on the interval [0, 1] does not converge, since the Riemann sum doesnt converge. Now, the case I'm slightly confused about is from looking at this proof https://imgur.com/a/sfwItRt . The proof is basically trying to show that the GL definition (dq f) and RL definition ([dqf]_{RL}) are equivalent. Both definitions extend the differentiation and integration operators to non-integer values. The GL definition is defined using left hand Riemann sum (in the imgur album). The sentence just above equation 3.3.1 says that if f(x) is bounded on a<y=<x then the definitions are equal, why does it not need to be bounded at a but bounded at x?

The set A= [;( t(1-t)^{\ n});] denotes the sequence of functions (from 0 to infinity). Is it totally bounded (or equicontinuous?) in C[0,1] where C[0,1] denotes all continuous functions in the given interval? or equicontinuous?

Hey MG! I'm growing an OG Kush in a spacebucket in FFOF, perlite, dolomite lime and feeding with the Go Box. Currently she's on her 17th day of flower. She's only in about 2.25 liters of soil. I went away for the weekend and came back earlier tonight to find her droopy and soft as hell. (~3.5 days without a watering) Her leaves had shrunken as well. Some of the main stems were even bending downward. I'm thinking she's approaching root boundedness thereby not retaining water as efficiently and drying out too quickly. I soaked her for a while and she seems to be gaining a bit of her turgor back. Does this sound accurate?

Should I repot knowing she's already on her 3rd week of flower, or stay more vigilant with my watering and ride it out til harvest? Has anyone else experienced this and has their plant fully recovered?

From wikipedia:

Let X be a Banach space and Y be a normed vector space. Suppose that F is a collection of continuous linear operators from X to Y. The uniform boundedness principle states that if for all x in X we have

Sup ||T(x)|| < infinity then sup ||T|| < infinity.

But if T is continuous => ||T|| < infinity, then how could sup ||T|| not be less than infinity ?

Also the course material from my school says that, if X banach Y normed etc. there are two possibilities, either

1. ||T|| < infinity with every T in F

OR

2. There is x in X, that sup ||Tx|| = infinity for every T in F

Are these definitions equivalent? I dont get it

I can't seem to find any good explanations of this concept online. Can someone give me a better understanding of what it means for a subset to be a combination of open, closed, bounded, and compact or none of these? I've seen the textbook definitions of these, but they don't make all that much sense.

also, please let me know if this could be better answered or found elsewhere. I checked r/learnmath and r/cheatatmathhomework but calc 3 seems to be the higher end of the questions there and cheatatmathhomework seemed to be mostly problem based. I'd like to understand the concept more than anything

For A_n = (ln(n^2 + 2n + 1))/(n+1), how would I determine the boundedness, monotonicity, and whether the sequence converges as n approaches infinity? And if it did converge, how would I find the limit?

"Show that every solution of the constant coefficient equation y''+a1y'+a2y=0 is bounded on[0<=x<infinity] if, and only if, the real parts of the characteristic polynomial are non positive and the roots with zero real part have multiplicity one."

I was thinking of the equation such that y''=0. In this case the roots of the characteristic polynomial are r=0 with multiplicity two. The solution would be phi = c1+xc2. However as x approaches infinity, this is not bounded and the premise of this question is not sound?

I am not sure if I am interpreting this question correctly (I have transcribed it exactly as in my text). I am first supposed to show that it is bounded if and only if the real parts of the roots are nonpositive and if the imaginary parts have multiplicity one, right? If that is the case r=0 with multiplicity two applies but the solution is unbounded.

I am currently looking over the proof to the Boundedness Theorem and was wondering what happened if you relaxed the closed bounded condition such that I=[a,b] is a bounded interval and f:I->R be continuous on I, then is f bounded on I? I think f wouldn't be bounded on I but I can't come up with a quick and dirty counterexample.

Another problem I was thinking about is if f(x) is continuous on a bounded interval I, then is f(x) uniformly continuous on I? I cannot think of a counterexample, but I can't prove it either. I know that lipschitz => UC but that isn't helping me.

I suspect both of these things are false, but am having trouble with finding an example that'd make them both false. But if it is true, how would you prove it?

Prove that an open ball in R^n is bounded.

The definition of boundedness of a subset A of R^n that we have been working with in class is that there exists an M in R such that for all u in A, ||u||<M. Starting with the open ball B of radius r about the point u, I know that for any point v in the ball, ||u-v|| < r. This is where I am stuck. I have tried manipulating this inequality to get ||v|| < something but I haven't had any luck. Can anyone offer me a hint?

I guess the concept didn't work

I don't want to step on anybody's toes here, but the amount of non-dad jokes here in this subreddit really annoys me. First of all, dad jokes CAN be NSFW, it clearly says so in the sub rules. Secondly, it doesn't automatically make it a dad joke if it's from a conversation between you and your child. Most importantly, the jokes that your CHILDREN tell YOU are not dad jokes. The point of a dad joke is that it's so cheesy only a dad who's trying to be funny would make such a joke. That's it. They are stupid plays on words, lame puns and so on. There has to be a clever pun or wordplay for it to be considered a dad joke.

Again, to all the fellow dads, I apologise if I'm sounding too harsh. But I just needed to get it off my chest.

Alot of great jokes get posted here! However just because you have a joke, doesn't mean it's a dad joke.

THIS IS NOT ABOUT NSFW, THIS IS ABOUT LONG JOKES, BLONDE JOKES, SEXUAL JOKES, KNOCK KNOCK JOKES, POLITICAL JOKES, ETC BEING POSTED IN A DAD JOKE SUB

Try telling these sexual jokes that get posted here, to your kid and see how your spouse likes it.. if that goes well, Try telling one of your friends kid about your sex life being like Coca cola, first it was normal, than light and now zero , and see if the parents are OK with you telling their kid the "dad joke"

I'm not even referencing the NSFW, I'm saying Dad jokes are corny, and sometimes painful, not sexual

So check out r/jokes for all types of jokes

r/unclejokes for dirty jokes

r/3amjokes for real weird and alot of OC

r/cleandadjokes If your really sick of seeing not dad jokes in r/dadjokes

Punchline !

Edit: this is not a post about NSFW , This is about jokes, knock knock jokes, blonde jokes, political jokes etc being posted in a dad joke sub

Edit 2: don't touch the thermostat