Abstract
The separable Hilbert spaces are the only class of abstract spaces for which the existence of a Schauder basis is warranted. It is indeed a classical fact that in every separable Hilbert space H there exists a total orthonormal sequence, hence an orthonormal basis for H. It follows that a basis for H is orthonormal if and only if it is normal and that such a basis is unconditional.
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Reference
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© 1969 Springer-Verlag Berlin · Heidelberg
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Marti, J.T. (1969). Bases for Hilbert Spaces. In: Introduction to the Theory of Bases. Springer Tracts in Natural Philosophy, vol 18. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-87140-5_6
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DOI: https://doi.org/10.1007/978-3-642-87140-5_6
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