We show that every finite-dimensional Alexandrov space X with curvature bounded from below embeds canonically into a product of an Alexandrov space with the same curvature bound and a Euclidean space such that each affine function on X comes from an affine function on the Euclidean space.
In this paper we give exact values of the best $n$-term approximation widths of diagonal operators between $ell_p(mathbb{N})$ and $ell_q(mathbb{N})$ with $0<p,qleq infty$. The result will be applied to obtain the asymptotic constants of best $n$-term approximation widths of embeddings of function spaces with mixed smoothness by trigonometric system.
We study functions of bounded variation (and sets of finite perimeter) on a convex open set $Omegasubseteq X$, $X$ being an infinite dimensional real Hilbert space. We relate the total variation of such functions, defined through an integration by parts formula, to the short-time behaviour of the semigroup associated with a perturbation of the Ornstein--Uhlenbeck operator.
We characterize the trace of magnetic Sobolev spaces defined in a half-space or in a smooth bounded domain in which the magnetic field $A$ is differentiable and its exterior derivative corresponding to the magnetic field $dA$ is bounded. In particular, we prove that, for $d ge 1$ and $p>1$, the trace of the magnetic Sobolev space $W^{1, p}_A(mathbb{R}^{d+1}_+)$ is exactly $W^{1-1/p, p}_{A^{shortparallel}}(mathbb{R}^d)$ where $A^{shortparallel}(x) =( A_1, dotsc, A_d)(x, 0)$ for $x in mathbb{R}^d$ with the convention $A = (A_1, dotsc, A_{d+1})$ when $A in C^1(overline{mathbb{R}^{d+1}_+}, mathbb{R}^{d+1})$. We also characterize fractional magnetic Sobolev spaces as interpolation spaces and give extension theorems from a half-space to the entire space.
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.