Let M be an N-function satisfying the Δ₂-condition, and let ω, φ be two other functions, with ω ≥ 0. We study Hardy-type inequalities ${\int }_{ℝ₊}M\left(\omega \left(x\right)\right|u\left(x\right)\left|\right)exp(-\phi \left(x\right))dx\le C{\int }_{ℝ₊}M\left(\right|u^{\text{'}}\left(x\right)\left|\right)exp(-\phi \left(x\right))dx$, where u belongs to some set of locally absolutely continuous functions containing $C{₀}^{\infty }\left(ℝ₊\right)$. We give sufficient conditions on the triple (ω,φ,M) for such inequalities to be valid for all u from a given set . The set may be smaller than the set of Hardy transforms. Bounds for constants are also given, yielding classical Hardy inequalities with best constants.

Let $X$ be an Archimedean Riesz space and $𝒫\left(X\right)$ its Boolean algebra of all band projections, and put ${𝒫}_e=\{Pe:P\in 𝒫\left(X\right)\}$ and ${ℬ}_e=\{x\in X:x\wedge (e-x)=0\}$, $e\in X^{+}$. $X$ is said to have Weak Freudenthal Property ($WFP$) provided that for every $e\in X^{+}$ the lattice $lin{𝒫}_e$ is order dense in the principal band $e^{dd}$. This notion is compared with strong and weak forms of Freudenthal spectral theorem in Archimedean Riesz spaces, studied by Veksler and Lavrič, respectively. $WFP$ is equivalent to $X^{+}$-denseness of ${𝒫}_e$ in ${ℬ}_e$ for every $e\in X^{+}$, and every Riesz space with sufficiently many projections...

We first introduce a notion of (a,b,c,d)-orthogonality in a normed linear space, which is a natural generalization of the classical isosceles and Pythagorean orthogonalities, and well known α- and (α,β)-orthogonalities. Then we characterize inner product spaces in several ways, among others, in terms of one orthogonality implying another orthogonality.

The main part of the paper is devoted to the problem of the existence of absolutely representing systems of exponentials with imaginary exponents in the spaces $C^{\infty }\left(G\right)$ and $C^{\infty }\left(K\right)$ of infinitely differentiable functions where G is an arbitrary domain in ${ℝ}^p$, p≥1, while K is a compact set in ${ℝ}^p$ with non-void interior K̇ such that $\overline{K}̇=K$. Moreover, absolutely representing systems of exponents in the space H(G) of functions analytic in an arbitrary domain $G\subseteq {ℂ}^p$ are also investigated.

Given a locally convex space (V,Γ), we find (all) the multiplications π on V (associative or not) such that the algebra A ≡ (V,π,Γ) becomes (i) A-convex, (ii) lm-convex.

Extensions of order bounded linear operators on an Archimedean vector lattice to its relatively uniform completion are considered and are applied to show that the multiplication in an Archimedean lattice ordered algebra can be extended, in a unique way, to its relatively uniform completion. This is applied to show, among other things, that any order bounded algebra homomorphism on a complex Archimedean almost $f$-algebra is a lattice homomorphism.

Let $F$ be a closed subset of ${ℝ}^n$ and let $P\left(x\right)$ denote the metric projection (closest point mapping) of $x\in {ℝ}^n$ onto $F$ in $l^p$-norm. A classical result of Asplund states that $P$ is (Fréchet) differentiable almost everywhere (a.e.) in ${ℝ}^n$ in the Euclidean case $p=2$. We consider the case $2<p<\infty$ and prove that the $i$th component $P_i\left(x\right)$ of $P\left(x\right)$ is differentiable a.e. if $P_i\left(x\right)\ne x_i$ and satisfies Hölder condition of order $1/(p-1)$ if $P_i\left(x\right)=x_i$.