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0o In this paper we study the connection of the rate of approximation of an element :c of a Banach space E by linear combinations of the elements of a given complete sequence 4e, [ and the possibility of representing this element as a series sum ~-~ a, e,.Even in a Hilbert space -H, for an arbitrary sequence ~,', 0 we can find a complete minimal system {e,!c H such that some element x~ is not representable as an even weakly convergent series ~a, e,, although for any n the distance of this element from the linear hull L~ of the first n. terms of system {e, ~ does not exceed ~. At the same time, for each linearly independent (but not necessarily Proposition 1. For any arbitrary sequence of numbers a~ -, o, in a separable Hilbert space H we can find a complete minimal sequence te,, K=4...tc H and an element ~ such that e~(~c)= e~ and if the c~ e, converges weakly, then its sum differs from ~. series Proof. Let t%, •=0, ~,... tc H be an orthonormalized basis in H. We set e~ --~ § 6, ~o, where t

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