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Let [itex]X[/itex] be a Hilbert (Banach) space, and spanned by a set [itex]S[/itex], say.

Let [itex]A[/itex] be linear bounded operator on X into itself.

Suppose that the operator is well known on S, that is

[itex]Aa_i=b_i[/itex] for all [itex]a_i\in S[/itex].

First, is this operator unique on X? if yes, can we find [itex]Aa[/itex], for general element a in X, in terms of [itex]b_i[/itex].

Thanks in advance

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# Operator on a set spaned the space

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