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Polynomial in finite-codimensional subspace of C[-1,1]

Source: Miklos Schweitzer 2023, Problem 9

March 7, 2024
polynomialanalysisFunctional AnalysisMiklos Schweitzerreal analysis

Problem Statement

Let C[−1,1]C[-1,1] be the space of continuous real functions on the interval [−1,1][-1,1] with the usual supremum norm, and let VV{} be a closed, finite-codimensional subspace of C[−1,1].C[-1,1]. Prove that there exists a polynomial p∈Vp\in V with norm at most one, which satisfies p′(0)>2023.p'(0)>2023.
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