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In this paper, equivalence constants between various polynomial norms are calculated. As an application, we also obtain sharp values of the Hardy--Littlewood constants for $2$-homogeneous polynomials on $ell_p^2$ spaces, $2<pleqinfty$ and lower estimates for polynomials of higher degrees.
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In this paper, we continue the investigation of topological properties of unbounded norm (un-)topology in normed lattices. We characterize separability and second countability of un-topology in terms of properties of the underlying normed lattice. We apply our results to prove that an order continuous Banach function space $X$ over a semi-finite measure space is separable if and only if it has a $sigma$-finite carrier and is separable with respect to the topology of local convergence in measure. We also address the question when a normed lattice is a normal space with respect to the un-topology.
Let $X$ and $Y$ be Banach spaces, let $mathcal{A}(X)$ stands for the algebra of approximable operators on $X$, and let $Pcolonmathcal{A}(X)to Y$ be an orthogonally additive, continuous $n$-homogeneous polynomial. If $X^*$ has the bounded approximation property, then we show that there exists a unique continuous linear map $Phicolonmathcal{A}(X)to Y$ such that $P(T)=Phi(T^n)$ for each $Tinmathcal{A}(X)$.
Various norms can be defined on a Krein space by choosing different underlying fundamental decompositions. Some estimates of norms on Krein spaces are discussed and few results in Bognars paper are generalized.
Given $n$ polynomials $p_1, dots, p_n$ of degree at most $n$ with $|p_i|_infty le 1$ for $i in [n]$, we show there exist signs $x_1, dots, x_n in {-1,1}$ so that [Big|sum_{i=1}^n x_i p_iBig|_infty < 30sqrt{n}, ] where $|p|_infty := sup_{|x| le 1} |p(x)|$. This result extends the Rudin-Shapiro sequence, which gives an upper bound of $O(sqrt{n})$ for the Chebyshev polynomials $T_1, dots, T_n$, and can be seen as a polynomial analogue of Spencers six standard deviations theorem.
Let $Bo(T,tau)$ be the Borel $sigma$-algebra generated by the topology $tau$ on $T$. In this paper we show that if $K$ is a Hausdorff compact space, then every subset of $K$ is a Borel set if, and only if, $$Bo(C^*(K),w^*)=Bo(C^*(K),|cdot|);$$ where $w^*$ denotes the weak-star topology and $|cdot|$ is the dual norm with respect to the sup-norm on the space of real-valued continuous functions $C(K)$. Furthermore we study the topological properties of the Hausdorff compact spaces $K$ such that every subset is a Borel set. In particular we show that, if the axiom of choice holds true, then $K$ is scattered.
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