Lethargy theorem

Let ${\displaystyle V_{1}\subset V_{2}\subset \ldots }$ be a strictly ascending sequence of finite-dimensional linear subspaces of a Banach space X, and let ${\displaystyle \epsilon _{1}\geq \epsilon _{2}\geq \ldots }$ 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 ${\displaystyle \epsilon _{i}}$.

• Bernstein's theorem (approximation theory)

• S.N. Bernstein (1938). "On the inverse problem of the theory of the best approximation of continuous functions". Sochinenya. II: 292–294.
• Elliott Ward Cheney (1982). Introduction to Approximation Theory (2nd ed.). American Mathematical Society. ISBN 978-0-8218-1374-4.
• Bauschke, Heinz H.; Burachik, Regina S.; Combettes, Patrick L.; Elser, Veit; Luke, D. Russell; Wolkowicz, Henry, eds. (2011). Fixed-Point Algorithms for Inverse Problems in Science and Engineering. Springer Optimization and Its Applications. Vol. 49. doi:10.1007/978-1-4419-9569-8. ISBN 9781441995681.
• Frank Deutsch; Hein Hundal (2010). "Slow convergence of sequences of linear operators I: almost arbitrarily slow convergence". Journal of Approximation Theory. 162 (9): 1701–1716. doi:10.1016/j.jat.2010.05.001. MR 2718892.
• Frank Deutsch; Hein Hundal (2010). "Slow convergence of sequences of linear operators II: arbitrarily slow convergence". Journal of Approximation Theory. 162 (9): 1717–1738. doi:10.1016/j.jat.2010.05.002. MR 2718893.
• Catalin Badea; Sophie Grivaux; Vladimir MÄuller (2011). "The Rate of Convergence in the Method of Alternating Projections". {{cite journal}}: Cite journal requires |journal= (help)
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