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In classical analysis, Lebesgue first proved that $mathbb{R}$ has the property that each Riemann integrable function from $[a,b]$ into $mathbb{R}$ is continuous almost everywhere. This property is named as the Lebesgue property. Though the Lebesgue property may be breakdown in many infinite dimensional spaces including Banach or quasi Banach spaces, to determine spaces having this property is still an interesting problem. In this paper, we study Riemann integration for vector-value functions in metrizable vector spaces and prove the fundamental theorems of calculus and primitives for continuous functions. Further we discovery that $mathbb{R}^{omega}$, the countable infinite product of $mathbb{R}$ with itself equipped with the product topology, is a metrizable vector space having the Lebesgue property and prove that $l^p(1<pleq+infty)$, as subspaces of $mathbb{R}^{omega}$, possess the Lebesgue property although they are Banach spaces having no such property.
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We investigate dynamical properties such as topological transitivity, (sequential) hypercyclicity, and chaos for backward shift operators associated to a Schauder basis on LF-spaces. As an application, we characterize these dynamical properties for weighted generalized backward shifts on Kothe coechelon sequence spaces $k_p((v^{(m)})_{minmathbb{N}})$ in terms of the defining sequence of weights $(v^{(m)})_{minmathbb{N}}$. We further discuss several examples and show that the annihilation operator from quantum mechanics is mixing, sequentially hypercyclic, chaotic, and topologically ergodic on $mathscr{S}(mathbb{R})$.
We continue the study dilation of linear maps on vector spaces introduced by Bhat, De, and Rakshit. This notion is a variant of vector space dilation introduced by Han, Larson, Liu, and Liu. We derive vector spa
Let $mathcal{D}$ be the class of radial weights on the unit disk which satisfy both forward and reverse doubling conditions. Let $g$ be an analytic function on the unit disk $mathbb{D}$. We characterize bounded and compact Volterra type integration operators [ J_{g}(f)(z)=int_{0}^{z}f(lambda)g(lambda)dlambda ] between weighted Bergman spaces $L_{a}^{p}(omega )$ induced by $mathcal{D}$ weights and Hardy spaces $H^{q}$ for $0<p,q<infty$.
This paper deals with a property which is equivalent to generalised-lushness for separable spaces. It thus may be seemed as a geometrical property of a Banach space which ensures the space to have the Mazur-Ulam property. We prove that if a Banach space $X$ enjoys this property if and only if $C(K,X)$ enjoys this property. We also show the same result holds for $L_infty(mu,X)$ and $L_1(mu,X)$.
We prove that for every Banach space $Y$, the Besov spaces of functions from the $n$-dimensional Euclidean space to $Y$ agree with suitable local approximation spaces with equivalent norms. In addition, we prove that the Sobolev spaces of type $q$ are continuously embedded in the Besov spaces of the same type if and only if $Y$ has martingale cotype $q$. We interpret this as an extension of earlier results of Xu (1998), and Martinez, Torrea and Xu (2006). These two results combined give the characterization that $Y$ admits an equivalent norm with modulus of convexity of power type $q$ if and only if weakly differentiable functions have good local approximations with polynomials.
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