# All Questions

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### Bernstein polynomial in Banach

For a continuous function $f : [0,1] \to R$, there exists a sequence of polynomial functions: $P_n(x)=\sum_{k=0}^n C^k_n x^k(1-x)^{n-k} f(\frac{k}{n})$ (Bernstein's polynomes) which converges ...
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### Show convexity of a function via inequalities

I am stuck with deriving the convexity of the function $$f(x) = \sqrt{1 + x^2}$$ from first principles, that is I would like to show that for any $x,y \in \mathbb R$ and $\lambda \in (0,1)$ we ...
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### Help understanding the proof of Trace Theorem given in Evans

I need help to understand the proof of the Trace Theorem given in Evans L.C. Partial differential equations (AMS, 1997): Asume $U$ is a bounded open set and that $\partial U$ is $C^1$. Then there ...
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### Prove this inequality with $a+b+c=3$

Let $a,b,c>0$,and $a+b+c=3$,show that $$\dfrac{a}{2b^3+c}+\dfrac{b}{2c^3+a}+\dfrac{c}{2a^3+b}\ge 1$$ such Use Cauchy-Schwarz inequality we have ...
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### Do the $2$ modulus $3$ can be $-1$ or just $2$?

I need to calculate $2$ modulus $3$ as $2<3$ then the answer should be $2$ but instead in a math problem they use it as $-1$. Is this possible? thanks
### What is an upper bound for $\|E(X|\mathcal{A})-E(X)\|$?
Let $X$ be a random element in a Banach space with norm $\|\cdot\|$, and $\mathcal{A}$ be a $\sigma$-algebra. What is an upper bound for $\|E(X|\mathcal{A})-E(X)\|$? Existing results: It has been ...
first please take a look at this: Given was a circle $c$ with center $A$ and ratio $r$, furthermore three lines $g$, $g1$, $g2$ with: $r = d(g, g1) = d(g, g2)$. Finally, two parabolas $p1$ (and ...