Continuous Rankin Bound for Hilbert and Banach Spaces

Let $(\mathrm{\Omega },\mu )$ be a measure space and $\{{\tau }_{\alpha }{\}}_{\alpha \in \mathrm{\Omega }}$ be a normalized continuous Bessel family for a real Hilbert space $\mathcal{H}$. If the diagonal $\mathrm{\Delta }:=\{(\alpha ,\alpha ):\alpha \in \mathrm{\Omega }\}$ is measurable in the measure space $\mathrm{\Omega }\times \mathrm{\Omega }$, then we show that \begin{align} (1) \quad\quad\quad\quad \sup _{α, β\in Ω, α\neq β}\langle τ_α, τ_β\rangle \geq \frac{-(μ\timesμ)(Δ)}{(μ\timesμ)((Ω\timesΩ)\setminusΔ)}. \end{align} We call Inequality (1) as continuous Rankin bound. It improves 76 years old result of Rankin [$\mathit{\text{Ann.}}\mathit{\text{of}}\mathit{\text{Math.,}}\mathit{\text{1947}}$]. It also answers one of the questions asked by K. M. Krishna in the paper [Continuous Welch bounds with applications, $\mathit{\text{Commun.}}\mathit{\text{Korean}}\mathit{\text{Math.}}\mathit{\text{Soc.,}}\mathit{\text{2023}}$]. We also derive Banach space version of Inequality (1).