Norming Approximate Orthogonality in Normed Linear Spaces

Abstract

We introduce and study the notion of \emph{norming approximate orthogonality}, a two-parameter generalization of Birkhoff--James orthogonality in normed linear spaces. For $\delta, \varepsilon \in [0,1)$ with $\varepsilon < (1-\delta)^2$, we say $x \nperp y$ in $X$ if there exists $f \in X^*$ with $|f(x)| \geq (1-\delta)\|f\|\|x\|$ and $|f(y)| \leq \frac{\varepsilon}{1-\delta}\|f\|\|y\|$, simultaneously relaxing both the norming condition on $x$ and the vanishing condition on $y$. It is proved that \[ x\nperp y \iff \|x+\lambda y\|\geq (1-\delta)\|x\|-\frac{\varepsilon}{1-\delta}\|\lambda y\|~\qquad \forall ~\text{scalars}~\lambda. \] This framework interpolates between two notions of approximate orthogonality in normed linear spaces due to Chmieli\'nski and Dragomir, and recovers three existing notions of orthogonality in extreme cases: exact Birkhoff--James orthogonality at $\delta = \varepsilon = 0$, the approximate orthogonality of Chmieli\'nski at $\delta = 0$, and the approximate orthogonality of Dragomir at $\varepsilon = 0$. A two-parameter proximity result generalizing Chmlie\'nski's characterization of $\perp_B^\varepsilon$ is established. The forward and converse implications are governed by the distinct thresholds $\frac{\varepsilon}{(1-\delta)^2}$ and $\frac{\varepsilon}{1-\delta}$, which collapse to $\varepsilon$ of Chmieli\'nski precisely when $\delta=0$, and the strictness of this gap is confirmed by counterexamples in $\ell_\infty^2$. A dual formulation of norming approximate orthogonality is established with a complete equivalence in the reflexive case. We apply our results to (vector-valued) continuous function spaces, which extends some earlier results and recovers few operator theoretical results with alternative proofs using measure theoretic techniques.