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 28 Evaluating $\lim\limits_{n\to\infty} \sum_{k=1}^{n^2} \frac{n}{n^2+k^2}$ 23 Infinite product of sine function 21 Compute: $\lim_{n\to\infty}\frac{{\left(1^1 \cdot 2^2 \cdot3^3\cdots n^n\right)}^\frac{1}{n^2}}{\sqrt{n}}$ 19 Inequality. $a^2+b^2+c^2 \geq a+b+c$ 18 if $f$ is entire and $|f(z)| \leq 1+|z|^{1/2}$, why must $f$ be constant?

# 36,502 Reputation

 +10 spectral projection of an element in a C*-algebra +10 Set of all compact operators $K(H)$ is the unique ideal in $B(H)$? +10 Prove that $\mathop {\lim }\limits_{(x,y,z) \to (0,0,0)} \left( {\frac{{{x^2}y - x{z^2}}}{{yz - {z^2}}}} \right)=0$ +10 Properties of reflexive Banach spaces

# 46 Questions

 37 Continuous projections in $\ell_1$ with norm $>1$ 19 Computing $\lim\limits_{n\to+\infty}n\int_{0}^{\pi/2}xf(x)\cos ^n xdx$ 17 Solving functional equation $f(x)^2+f(y)^2=f(x+y)(f(f(x))+f(y))$ 15 Minmax problem for polygons 14 Is it true that $\mathbb{R}$ and $\mathbb{R}^2$ are isomorphic as abelian groups?

# 179 Tags

 882 functional-analysis × 430 226 banach-spaces × 115 434 real-analysis × 147 190 limits × 39 432 calculus × 90 173 operator-theory × 91 239 sequences-and-series × 55 150 measure-theory × 65 238 integration × 57 137 analysis × 56

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