Related papers: Proof mining in $L^p$ spaces
In this paper, we study Auerbach basis of the Banach spaces $l^n_p$. We provide a complete classification of the spaces in terms of the cardinality of their bases. We also give a complete description of these bases for $l^3_p$ ($l^2_p$ is…
The usual examples of Bergman spaces consist of the closure of an algebra of holomorphic functions on a domain. One can also take the real part of such functions, but essentially one is looking at the same object. In this paper the author…
Higher-order constructs extend the expressiveness of first-order (Constraint) Logic Programming ((C)LP) both syntactically and semantically. At the same time assertions have been in use for some time in (C)LP systems helping programmers…
Formal deductive systems are very common in computer science. They are used to represent logics, programming languages, and security systems. Moreover, writing programs that manipulate them and that reason about them is important and…
We analyze a proof of Bruck to obtain an explicit rate of asymptotic regularity for Ces\`aro means in uniformly convex Banach spaces. Our rate will only depend on a norm bound and a modulus $\eta$ of uniform convexity. One ingredient for…
Given an infinite matrix $M=(m_{nk})$ we study a family of sequence spaces $\ell_M^p$ associated with it. When equipped with a suitable norm $\|\cdot\|_{M,p}$ we prove some basic properties of the Banach spaces of sequences…
In various provers and deductive verification tools, logical transformations are used extensively in order to reduce a proof task into a number of simpler tasks. Logical transformations are often part of the trusted base of such tools. In…
We address generating theorems from a given set of axioms, without proof goal, aiming at value from a mathematical point of view or as lemmas for automated proving. As benchmark, we convert a fragment of the Metamath database set.mm. Our…
Motivated by the Lyapunov convexity theorem in infinite dimensions, we extend the convexity of the integral of a decomposable set to separable Banach spaces under the strengthened notion of nonatomicity of measure spaces, called…
We provide a sufficient condition for the sum of a finite number of complemented subspaces of a Banach space to be complemented. Under this condition a formula for a projection onto the sum is given. We also show that the condition is sharp…
We define a p-norm in the context of quantum random variables, measurable operator-valued functions with respect to a positive operator-valued measure. This norm leads to a operator-valued L^p space that is shown to be complete. Various…
The linear isometries between weighted Banach spaces of continuous functions are considered. Some of well known theorems on isometries between spaces of continuous functions are proved and stated, but all they are in an appropriate form. In…
We study the interplay between Banach space theory and theory of analytic P-ideals. Applying the observation that, up to isomorphism, all Banach spaces with unconditional bases can be constructed in a way very similar to the construction of…
Large Language Models (LLMs) are often used as automated judges to evaluate text, but their effectiveness can be hindered by various unintentional biases. We propose using linear classifying probes, trained by leveraging differences between…
We consider structured optimisation problems defined in terms of the sum of a smooth and convex function, and a proper, l.s.c., convex (typically non-smooth) one in reflexive variable exponent Lebesgue spaces $L_{p(\cdot)}(\Omega)$. Due to…
Let N and M be von Neumann algebras. It is proved that L^p(N) does not Banach embed in L^p(M) for N infinite, M finite, 1 < or = p < 2. The following considerably stronger result is obtained (which implies this, since the Schatten p-class…
We describe all metric spaces that have sufficently many affine functions. As an application we obtain a metric characterization of linear-convex subsets of Banach spaces.
We define spatial $L^p$ AF algebras for $p \in [1, \infty) \setminus \{ 2 \}$, and prove the following analog of the Elliott AF algebra classification theorem. If $A$ and $B$ are spatial $L^p$ AF algebras, then the following are equivalent:…
Complex black-box machine learning models are regularly used in critical decision-making domains. This has given rise to several calls for algorithmic explainability. Many explanation algorithms proposed in literature assign importance to…
Real-valued logics underlie an increasing number of neuro-symbolic approaches, though typically their logical inference capabilities are characterized only qualitatively. We provide foundations for establishing the correctness and power of…