real analysis – compactness of complete sequence spaces

Originally, my problem was to find a set that is limited but not completely bounded in a non-complete metric space. Then I wanted to think $ (Fin, d _ { infty}) $, from where $ Fin $ is the set of all finite consequences, and $ d _ { infty} $ is the highest metric. Actually, I am looking for a subgroup of $ Fin $ that is limited and not completely limited.

Case 1: If $ (c_0, d _ { infty}) $ is compact, where $ c_0 $ is the set of all null sequences, that is, since the completion of $ Fin $ is this room, $ Fin $ must be completely limited. Therefore, each subset of it is also totally limited. Therefore, this is a bad example of finding an amount that is limited and not completely limited. Then I became curious. What would happen in the general case?

Given sentence $ A $ If the completion in a non-complete space is compact, there is no subset $ V $ from $ A $ so that $ V $ is not completely limited. If the completion is not compact, I might find such a subset. Therefore, I would like to know which of the following complete sequence spaces are compact:

1)$ (c_0, d _ { infty}) $,
2)$ (c, d _ { infty}) $, from where $ c $ is the set of convergent sequences.
3)$ (l _ { infty}, d _ { infty}) $, from where $ l _ { infty} $ is the set of all limited consequences.
4)$ (l_p, d_p) $, from where $ l_p $ is the set of all "p-summable" sequences.
5)$ ( omega, d_ {Fr}) $, from where $ omega $ is the set of all episodes and $ d_ {Fr} $ is the Frechet metric.

In addition, is there a problem with my reasoning? If so, can you explain that to me?

Thanks for any help.

real analysis – proof of the continuity of the Lebesgue in the monotone convergence theorem

I try to prove the following statement: Let $ f $ to be a non-negative measurable function. We define $ F (x) = int _ {- infty} ^ x {f (u) you} $, Show that $ F $ is continuous using the monotone convergence theorem.

I started writing the integral as:

$$ F (x_0) = int _ {- infty} ^ {x_0} {f (u) du} = int {f (u) cdot chi _ {(- infty, x_0)} (u ) you} $$

and tried to prove the claim with Heine's criterion: $ F (x) $ is always on $ x_0 $ if and only if for each sequence $ {x_n } _ {n = 1} ^ infty $ so that $ x_n rightarrow x_0 $. $ F (x_n) rightarrow F (x_0) $,

I have managed to prove this for monotone sequences, i. $ x_n uparrow x_0 $ and $ x_n downarrow x_0 $However, using the monotone convergence theorem, I got stuck trying to prove this for a general sequence.

Show that there is only one maximum solution to a differential equation defined for all real numbers.

I try to solve a problem with ordinary differential equations.

I am given this ODE: $ y = = y ^ 2 + ( sin (xy)) ^ 2 $,

I am asked to prove that the null function is the only maximal solution that is defined for all reals of this ODE.

These are a few things that I thought about:

  • Since the solution must be unique for an IVP (Picard's Theorem) and Y = 0 is a solution, all other solutions must have a constant sign, since they can not overlap with this solution.
  • The solutions increase functions because their derivative is positive.

However, I can not prove that the rest of the maximum solutions are not defined for all realities. By the way, I do not think that I have to solve the equation to answer this problem. I think it's more of a theoretical exercise.

If someone could give me a hint, that would be very helpful.

Many thanks.

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Matrices – differences between combinations of real numbers

I would like to ask what the difference is

$ mathbb {R} ^ n, mathbb {R} ^ {n times n}, mathbb {R} ^ {2n} $?

I try to understand what the difference is, because for $ mathbb {R} ^ n $ They place vectors as coulums in a matrix.

Example of a $ mathbb {R} ^ 2 $ as I understand it is:

$ vec {a} ^ T = (1,2), vec {b} ^ T = (3,4) Rightarrow underline {A} = left ( begin {array} {c} 1 & 3 \ 2 & 4 end {array} right) $

but for just a tuple of numbers, I get that for A, which is not a matrix:

$ A = {(1,3), (1,4), (2,3), (2,4) } $

I am a bit confused what the rest of them would be and how to take the position for matrices.
Thanks for any help to understand the differences. Examples would be alright.

br

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Is the latency of AWS / GCP / Azure Singapore intolerable? u-buy.vn seems to use AWS Singapore. I VPN to Singapore when in Hanoi and I think the latency is about 30ms out of memory. I do not believe that there will ever be real cloud providers in countries like Vietnam, Malaysia, Laos and Thailand until the elite's thinking is fundamentally changing (xenophobia towards foreign companies and extreme interest in monopoly protection) – in the meantime Singapore, Hong Kong and Taiwan are industrialized countries where the big 3's play a role. Cloudflare and Azure have edge nodes in Vietnam if they are able to use their CDN. If you're a little googling, I'm shocked to learn Cloudfront (AWS), and GCP CDN has no edge node in Vietnam.

They also have Linode / Vultr / DigitalOcean in the SGP when the big 3 are outside the price range.

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