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"If you want to build a ship, don't drum up the men to gather wood, divide the work, and give orders. Instead, teach them to yearn for the vast and endless sea." (Antoine de Saint Exupéry)

"Borders? I have never seen one. But I have heard they exist in the minds of some people." (Thor Heyerdahl)

"We are not now that strength which in old days \ moved earth and heaven, that which we are, we are, \ One equal temper of heroic hearts,\ Made weak by time and fate, but strong in will \ To strive, to seek, to find, and not to yield." (Alfred Tennyson)


Sep
15
awarded  Self-Learner
Aug
9
reviewed Approve suggested edit on How to solve this IVP?
Aug
9
comment Do the radii of a family of nested balls (in a Banach space) converge?
That's interesting and surprising for me, many thanks. Can I ask you which is the problem for $\mathbb Q_p$, thus? Why does Henning Makholm's proof below does not apply in thi case? I am not very familiar with $p$-adic numbers, to be honest. Thanks again for your comments.
Aug
9
comment Do the radii of a family of nested balls (in a Banach space) converge?
@you-sir-33433 Sure, thanks for pointing it out; but if you are in a Banach space and the balls are nested then necessarily the intersection is not empty.
Aug
9
comment Outer Measure is not Finite Additive
You may find this interesting. math.stackexchange.com/questions/878962/…
Aug
9
accepted Do the radii of a family of nested balls (in a Banach space) converge?
Aug
9
comment Do the radii of a family of nested balls (in a Banach space) converge?
Oh, I see your point. Thank you very much for the kind and fast reply.
Aug
9
asked Do the radii of a family of nested balls (in a Banach space) converge?
Aug
9
comment Nested sequences of balls in a Banach space
Sorry for this comment which is probably stupid but I am wondering how you would prove that $B_r(x) \subset B_s(x)$ implies $r \le s$. What about these examples: math.stackexchange.com/questions/734248/… ? Thanks.
Aug
8
accepted A strictly convex function defines an implicit function with non-positive second derivative
Aug
7
revised Jordan Measure and Lebesgue Measure
edited title
Aug
6
comment How do I approach the problems asking about uniqueness
@user86418 Oh, sorry: you are perfectly right, I have been stupid. Thanks for pointing it out, now it makes sense.
Aug
6
comment How do I approach the problems asking about uniqueness
The solution of the first problem is straigthforward if we do not ask for periodicity (just take $C(\mathbb R)$ with the sup norm and verify that the "obvius" map is 1/2-Lipschitz: then apply Banach-Caccioppoli fixed point theorem). I am wondering what changes adding periodicity condition: the space of continuous periodic functions is still complete under the sup norm hence I think that the same argument applies.
Aug
5
revised A strictly convex function defines an implicit function with non-positive second derivative
Added a possible solution to new doubt
Aug
5
revised A strictly convex function defines an implicit function with non-positive second derivative
Added a new doubt
Aug
4
comment A strictly convex function defines an implicit function with non-positive second derivative
This is nice, thank you for your answer. I have been very stupid because I did not notice that. Thanks again. By the way, what do you think about the first point? Is it correct? I am now doubtful because I have proved only that the difference quotients are $>0$, maybe when we pass to the limit we get $0$... Thanks.
Aug
4
asked A strictly convex function defines an implicit function with non-positive second derivative
Jul
28
comment Finite additivity in outer measure
You are welcome.
Jul
26
accepted About the $\lim_{n \to+\infty} \frac{1}{n}\int_0^1 \log(1+e^{nf(x)})\,dx$ (Rudin's exercise)
Jul
26
comment About the $\lim_{n \to+\infty} \frac{1}{n}\int_0^1 \log(1+e^{nf(x)})\,dx$ (Rudin's exercise)
Oh, yes I see: you are right. Thanks, Davide, for your precious answer.