Little-o multiscale coefficient space M0(w)

Use the wavelet coefficient levels $(l,k)$ and multiplicities $\mathrm{\mid }{K}_l\mathrm{\mid }=2^l$ from Logarithmic negative Sobolev spaces for white noise. The same coefficient-space definition applies to the Haar system used for type-H histograms: detail levels $l\ge 0$ have $2^l$ coefficients, and the single scaling coefficient is an additional finite coarsest coordinate, included in the norm with weight $1$. Finite projections always retain that scaling coordinate; this finite convention does not alter any tail assertion. Let $w=(w_l)$ be nondecreasing with $w_l\ge 1$. Define $M(w)=\{x=(x_{lk}):\mathrm{\parallel }x{\mathrm{\parallel }}_{M(w)}:={\sup }_l\frac{{\max }_{k\in K_l}\mathrm{\mid }{x}_{lk}\mathrm{\mid }}{w_l}<\mathrm{\infty }\}$ and its closed little-o subspace $M_0(w)=\{x\in M(w):\frac{{\max }_{k\in K_l}\mathrm{\mid }{x}_{lk}\mathrm{\mid }}{w_l}⟶0\text{ as }l\to \mathrm{\infty }\},$ with the same norm. Call $w$ admissible when $w_l\mathrm{/}\sqrt{l}\to \mathrm{\infty }$. The finite coarsest level is included in the norm and does not affect this limit. Elements are coefficient sequences; in particular, they need not be $L^2$ functions. The multiscale coefficient space is separable and has vanishing projection tails ↗ proves that $M_0(w)$ is a separable Banach space, that $L^2$ coefficient sequences embed in it, and that finite-level projections converge in its norm.