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An unconventional orthonormal basis provides an unexpected counterexample. (English) Zbl 0820.46008

The authors pose the following two questions and answer the first with “yes” and the second – using the affirmative answer of the first – with “no”.
Question 1. Does there exist an orthonormal basis in \(L^ 2[a, b]\) containing only one discontinuous function?
Question 2. Let \(S\) be a real inner product space and let \(H\) be the real Hilbert space completion of \(S\). Let \(T: S\to S\) be a compact self- adjoint operator. Let \(\overline T: H\to H\) be the continuous extension of \(T\) to \(H\). If \(T\) is injective, does it follow that \(\overline T\) must also be injective.

MSC:

46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
47B07 Linear operators defined by compactness properties
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