Lethargy theorem

In mathematics, a lethargy theorem is a statement about the distance of points in a metric space from members of a sequence of subspaces; one application in numerical analysis is to approximation theory, the difficulty of approximating general functions by functions of special form, such as polynomials.

Bernstein's lethargy theorem. Let $V_{1}\subset V_{2}\subset \ldots$ be a strictly ascending sequence sequence of finite-dimensional linear subspaces of a Banach space X, and let $\epsilon _{1}\geq \epsilon _{2}\geq \ldots$ be a decreasing sequence of real numbers tending to zero. Then there exists a point x in X such that the distance of x to V_i is exactly $\epsilon _{i}$.

References

• S.N. Bernstein (1938). "On the inverse problem of the theory of of the best approximation of continuous functions". Sochinenya. II: 292–294.