Let U, V be two symmetric convex bodies in $ℝ^n$ and |U|, |V| their n-dimensional volumes. It is proved that there exist vectors $u_1,...,u_n ∈ U$ such that, for each choice of signs $ε_1,...,ε_n = ± 1$, one has $ε_1 u_1 + ... + ε_n u_n ∉ rV$ where $r = (2πe^2)^{-1/2} n^{1/2}(|U|/|V|)^{1/n}$. Hence it is deduced that if a metrizable locally convex space is not nuclear, then it contains a null sequence $(u_n)$ such that the series $∑_{n = 1}^∞ ε_n u_{π(n)}$ is divergent for any choice of signs $ε_n = ± 1$ and any permutation π of indices.
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Nuclear groups form a class of abelian topological groups which contains LCA groups and nuclear locally convex spaces, and is closed with respect to certain natural operations. In nuclear locally convex spaces, weakly summable families are strongly summable, and strongly summable are absolutely summable. It is shown that these theorems can be generalized in a natural way to nuclear groups.
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The paper deals with the approximation by polynomials with integer coefficients in $L_p(0,1)$, 1 ≤ p ≤ ∞. Let $P_{n,r}$ be the space of polynomials of degree ≤ n which are divisible by the polynomial $x^r(1-x)^r$, r ≥ 0, and let $P_{n,r}^ℤ ⊂ P_{n,r}$ be the set of polynomials with integer coefficients. Let $μ(P_{n,r}^ℤ;L_p)$ be the maximal distance of elements of $P_{n,r}$ from $P_{n,r}^ℤ$ in $L_p(0,1)$. We give rather precise quantitative estimates of $μ(P_{n,r}^ℤ;L₂)$ for n ≳ 6r. Then we obtain similar, somewhat less precise, estimates of $μ(P_{n,r}^ℤ;L_p)$ for p ≠ 2. It follows that $μ(P_{n,r}^ℤ;L_p) ≍ n^{-2r-2/p}$ as n → ∞. The results partially improve those of Trigub [Izv. Akad. Nauk SSSR Ser. Mat. 26 (1962)].