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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.

Edward Odell worked on the geometry of Banach spaces. With Thomas Schlumprecht, he solved the distortion problem for the classical infinite-dimensional Hilbert space by proving it arbitrarily distortable.^[ref]

An equivalent norm can preserve the underlying vector-space topology yet stretch unit vectors very differently inside every infinite-dimensional subspace. For any prescribed ratio greater than one, their construction forces such variation. This distinguishes topological equivalence of norms from uniform control of their geometry on large subspaces.^[ref]

Thomas Schlumprecht studies the theory of Banach spaces. His joint work with Edward Odell proved arbitrary distortion of infinite-dimensional Hilbert space.^[ref]

For each chosen distortion threshold, one constructs an equivalent norm whose variation cannot be avoided by passing to any infinite-dimensional subspace. The same threshold need not describe a single norm for every possible ratio: the norm is chosen after the desired ratio. This quantifier order is central to the result and to the notion of bounded distortion of a Banach space.^[ref]