### Questions (10)

 3 Infinite composition of function that has a fixed point 2 Given an averaging operator $A: l_p \rightarrow l_p$, why $A$ is not compact 2 Group action on a set that has smaller cardinality than group's one 1 Easy way to extend continous functional 1 Prove that a certain closed subspace of $C[0,1]$ has finite dimension

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 2 f is a absolutely continuous function on $[a,b]$ prove that $①\int_a^b\vert f'(t) \vert dt=V_{a}^b$ 1 Prove $\sum\limits_{n=1}^{\infty} a_{f(n)} = \dfrac{3}{2} \ln(2)$ 1 Show that $\lim_{n \rightarrow \infty}\sqrt[n]{n}=1$ using Bernoulli inequality 1 Proving a bounded and continuous function $f: [a, b] \rightarrow \mathbb{R}$ assumes its maximum value on $[a, b].$ 1 Isometric embedding of $l_\infty$ to $L_\infty[0;1]$

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 6 real-analysis × 7 1 limits × 2 1 functional-analysis × 7 1 sequences-and-series 1 isometry × 2 0 lp-spaces × 3 1 convergence × 2 0 lebesgue-measure 1 calculus × 2 0 compact-operators

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