  For any sequence of positive numbers $(\varepsilon_n)_{n=1}^\infty$ such that
$\sum_{n=1}^\infty \varepsilon_n = \infty$ we provide an explicit simple
construction of $(1+\varepsilon_n)$-bounded Markushevich basis in a separable
Hilbert space which is not strong, or, in other terminology, is not hereditary
complete; this condition on the sequence $(\varepsilon_n)_{n=1}^\infty$ is
sharp. Using a finite-dimensional version of this construction, Dvoretzky's
theorem and a construction of Vershynin, we conclude that in any Banach space
for any sequence of positive numbers $(\varepsilon_n)_{n=1}^\infty$ such that
$\sum_{n=1}^\infty \varepsilon_n^2 = \infty$ there exists a
$(1+\varepsilon_n)$-bounded Markushevich basis which is not a Schauder basis
after any permutation of its elements.
