Every frame in Hilbert space contains a subsequence equivalent to an orthogonal basis. If a frame is n-dimensional then this subsequence has length (1 - ε)n. On the other hand, there is a frame which does not contain bases with brackets.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm145-3-1,
author = {R. Vershynin},
title = {Subsequences of frames},
journal = {Studia Mathematica},
volume = {147},
year = {2001},
pages = {185-197},
zbl = {0987.46028},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm145-3-1}
}
R. Vershynin. Subsequences of frames. Studia Mathematica, Tome 147 (2001) pp. 185-197. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm145-3-1/