 Bounded distortion of a Banach space

For equivalent norms $q$ and $\Vert \cdot \Vert$ on an infinite-dimensional closed linear subspace $Y$, define its relative distortion on a further infinite-dimensional closed linear subspace $Z\subseteq Y$ byThe equivalent norms condition makes the denominator positive and the numerator finite. A Banach space has bounded distortion if one constant $D<\infty$ works for every such $Y$ and every $q$ among the equivalent norms on $Y$: there is a further infinite-dimensional closed linear subspace $Z\subseteq Y$ with ${\mathrm{dist}}_Z(q,\Vert \cdot \Vert )\le D$. The constant is uniform over renormings and starting subspaces; $Z$ may depend on both.

This convention is expressed by saying that, for some finite distortion threshold, no subspace is distortable beyond that threshold. A block-subspace formulation additionally assumes a specified Schauder basis and tests further block subspaces; the basis must be stated rather than silently assumed.^[ref]

The ratio is unchanged by multiplying $q$ by a positive scalar. Equivalently, on the selected $Z$ there is $a>0$ such that $a\Vert z\Vert \le q(z)\le aD\Vert z\Vert$. For $q=100\Vert \cdot \Vert$, the distortion is exactly one, not one hundred. Allowing that rescaling is essential: distortion compares relative geometry rather than the arbitrary unit of length.