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 $\hat{I}´,\epsilon \in [0,1)$ with $\epsilon <(1-\hat{I}´{)}^2$, we say $x \nperp y$ in $X$ if there exists $f\in X^{\ast }$ with $|f(x)|\ge (1-\hat{I}´)\Vert f\Vert \Vert x\Vert$ and $|f(y)|\le \frac{\epsilon }{1-\hat{I}´}\Vert f\Vert \Vert y\Vert$, simultaneously relaxing both the norming condition on $x$ and the vanishing condition on $y$. It is proved that \[ x\nperp y \iff \|x+λy\|\geq (1-δ)\|x\|-\frac{\varepsilon}{1-δ}\|λy\|~\qquad \forall ~\text{scalars}~λ. \] This framework interpolates between two notions of approximate orthogonality in normed linear spaces due to Chmieliński and Dragomir, and recovers three existing notions of orthogonality in extreme cases: exact Birkhoff--James orthogonality at $\hat{I}´=\epsilon =0$, the approximate orthogonality of Chmieliński at $\hat{I}´=0$, and the approximate orthogonality of Dragomir at $\epsilon =0$. A two-parameter proximity result generalizing Chmlieński's characterization of ${\perp }_B^{\epsilon }$ is established. The forward and converse implications are governed by the distinct thresholds $\frac{\epsilon }{(1-\hat{I}´{)}^2}$ and $\frac{\epsilon }{1-\hat{I}´}$, which collapse to $\epsilon$ of Chmieliński precisely when $\hat{I}´=0$, 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.

20 Pages