Abstract:
The question of the linearity of the set of sums of the function series $\sum^\infty_{n=1}\varphi_n(x)$ is investigated. It is shown that the requirement $\sum^\infty_{n=1}\|\varphi_n\|^p_{L_p}<\infty$ in the theorem of Kadec ensuring the linearity of the set of sums of a series in the spaces $L_p(0,1)$ with $1\leqslant p\leqslant2$ is definitive. In §2 it is shown that no nontrivial requirement on the norms of the functions of the series or on their absolute values can be sufficient for the linearity of the set of sums of the series in the space $C[a,b]$.
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