tchao
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Sequence in $\ell^2$ converget to $0$ if it is summable
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5 votes

No, this is not true. Take the series $\nu_n=\frac{1}{n}$.

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Finding the expected value of of $\int_0^s \sqrt{t+B_t^2}dB_t$?
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3 votes

I think you already wrote down everything you need to solve your problem. About measurability. I assume that we are speaking about the filtration $\mathcal F_t$ that is generated by the given ...

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Square summable sequences of complex numbers is simply summable?
2 votes

Yes, it is true. Since the series $\sum_{n=1}^\infty |a_n|^2$ converges to zero, the sequence $S_N:=\sum_{n=1}^N |a_n|^2$ must converge to zero. But we have: $S_N\geqslant S_{N-1}\geqslant 0$, ...

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Compactness and epsilon delta
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2 votes

The statement is true if your function is continuous. Any continuous real-valued function on a compact set takes its infinum (and supremum) at some point in this set. If $g(z)$ is continuous, $|g(z)|$ ...

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Corollary to Bolzano weierstrass theorem
1 votes

This sentence is the negation of the statement that $a_n\to -\infty$. If $a_n\to -\infty$, then for every $\beta\in\mathbb R$, there exists $n_0\in\mathbb N$ such that for all $n>n_0$ we have $a_n&...

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Show that $|a(u,v)|\leq M\|u\|\|v\|$ and $|a(v,v)|\geq \nu\|v\|^2$ on $H_0^1(\Omega )$.
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1 votes

1) Yes. 2) Note that on the right-hand side of your estimate you have $\lVert \nabla u \rVert_{L^2}^2\leq \lVert u \rVert_{H_0^1}^2$ and $\lVert u \rVert_{L^2}^2\leq \lVert u \rVert_{H_0^1}^2$, see ...

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Cartesian product of subsets $A$ and $B$ of $X$ and $Y$ expressed as $X\times Y - (...)$
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1 votes

The formula from your comment is correct. Observe that for sets $C, D, E, F$ we have $$ C-(D\cup E)=(C-D)\cap (C-E),$$ $$(C-D)\times E=C\times E - D\times E,$$ and $$(C\times D)\cap (E\times F)=(C\...

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Why is $\lim_{t\to 1}\frac{t^n-1}{t^m-1}=\frac{n}{m}$
1 votes

Because $$\frac{(t-1)(t^{n-1}+...+1)}{(t-1)(t^{m-1}+...+1)}=\frac{t^{n-1}+...+1}{t^{m-1}+...+1},$$ and all powers of $t$ converge to $1$ as $t$ goes to $1$. There are $n$ summands in numerator (the $n-...

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When is $\mathbb P(A\cap B)$ minimale and maximale?
1 votes

The probability $\mathbb P(A\cap B)$ is minimal (actually, zero) if $A$ and $B$ are complementary events, like $A$ having heads and $B$ having tails when throwing a coin once. Note that these events ...

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How to define $\mathrm{E}[X|Y]$ when $\mathrm{E}[X|Y=y]=\sum_{x \in \mathcal{X}}x \cdot P(X=x|Y=y) $
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1 votes

It is $$E[X|Y]=\sum_{y \in \mathcal Y}\Big[\sum_{x \in \mathcal X}x\cdot P(X=x|Y=y)\Big]\cdot \chi_y(Y),$$ where $\chi_y(Y)=1$ if $Y$ takes the value $y$ and zero otherwise. In that way, you indeed ...

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Find $\liminf$ and $\limsup$ of sequence
1 votes

We can derive separate formulas for $(x_{2k})$ and $(x_{2k+1})$. For even indices we have: $x_2=0$ and $x_{2k+2}=\frac{1}{2}+\frac{x_{2k-2}}{2}$ for $k>1$. One can easily show by induction that ...

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Relationship Between Diagonalizability and Jordan Form
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1 votes

I would rather say that if $A$ is diagonalizable, it is also "jordanizable", and its Jordan form $J$ is exactly $D$, the corresponding diagonal matrix. Think of every diagonal element of $D$ as of a ...

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How one can calculate $\mathbb{E}[\sin(5B_t)]$?
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0 votes

Recall that $B_t$ is normally distributed with mean $0$ and variance $t$. Then by the "law of the unconscious statistician", $$\mathbb E(\sin(5B_t))=\frac{1}{\sqrt{2\pi t}}\int_{-\infty}^{\infty}\sin(...

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How to prove $E|X|^p \le E|X+Y|^p$ given certain conditions?
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0 votes

With the hints of Sergei Golovan: Consider conditional expectation $ E(|X+Y|^p|\ X)$ and apply Jensen's inequality to it: $$E(|X+Y|^p|\ X )\geq |E(X+Y|\ X)|^p. $$ Moreover, since $X$ and $Y$ are ...

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Continuous random variable - coefficients and distributive function
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(a) Your value of $a$ seems to be incorrect, but you use correct relations. (b) The distribution function is an increasing function such that $\lim_{t\to -\infty }F(t)=0$, $\lim_{t\to +\infty }F(t)=...

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Convert second order PDE $u_{tt} = u_{xx} + u$ to a system of first order PDE's
0 votes

You should introduce two more functions $v$ and $w$, so that your system of three equations does indeed contain three unknown functions. Then you specify that $v$ and $w$ are, respectively, the first ...

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