2

Here $c$ means the cardinal number of the continuum.

This is an exercise (4.8.7) in the book of Avner Friedman, 'foundations of modern analysis'.

I am a little bit surprised.

Just consider the $l^2$ Hilbert space. It is well-known that it has a basis $\{e_n = (0,0,\ldots, 1, 0,\ldots )\}$, which is countable. As I understand, $l^2$ is a Banach space, a Hilbert space actually, whose dimension is countably infinite.

In which place am I wrong?

John
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    The basis which you mention of $l^2$ is not a basis in the same sense of the exercise you refer. In the exercise, the author probably means the dimension in the purely algebraic sense. That basis of $l^2$ is not a basis in this sense, it is a basis in the sense that the closure of its span is $l^2$. It seems that the answers in this question address the problem you want. – Aloizio Macedo Oct 18 '19 at 02:24
  • Isn't ${e_i} $ a Schauder basis? IIRC, there's a theorem which says that a Banach space cannot have a Hamel basis of countable cardinality. – cqfd Oct 18 '19 at 02:24

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