We study the relation between different spaces of vector-valued polynomials and analytic functions over dual-isomorphic Banach spaces. Under conditions of regularity on$E$and$F$, we show that the spaces of$X$-valued$n$-homogeneous polynomials and analytic functions of bounded type on$E$and$F$are isomorphic whenever$X$is a dual space. Also, we prove that many of the usual subspaces of polynomials and analytic functions on$E$and$F$are isomorphic without conditions on the involved spaces.

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We prove that $$ \mathop{ \lim \inf}\limits_{n \rightarrow \infty} \frac{p_{n+1}-p_{n}}{\sqrt{\log p_{n}} \left(\log \log p_{n}\right)^{2}}< \infty, $$ where$p$_{$n$}denotes the$n$th prime. Since on average$p$_{$n$+1}−$p$_{$n$}is asymptotically log_{$n$}, this shows that we can always find pairs of primes much closer together than the average. We actually prove a more general result concerning the set of values taken on by the differences$p$−$p$′ between primes which includes the small gap result above.

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No abstract uploaded!