Geodesic
A~nowhere dense space of linear superpositions of functions of several variables
Izvestiya. Mathematics , Tome 6 (1972) no. 4, pp. 807-837

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Let $\mathbf I^3$ be the unit cube of three-dimensional space $R^3$, and let $\Phi_i(x)$, $i=1,\dots,n$, be mappings $\Phi_i\colon\mathbf I^3\to R^2$ of class $C_2$. We prove that the set of functions $F(x)$ on $\mathbf I^3$ which can be represented in the form $$ F(x)=\sum_{i=1}^n(\chi_i\circ\Phi_i)(x), $$ where the $\chi_i(u_1,u_2)$ are arbitrary continuous functions, $\chi_i\colon R^2\to R$, is nowhere dense in $\mathscr L_2(\mathbf I^3)$. 
@article{IM2_1972_6_4_a7,
     author = {B. L. Fridman},
     title = {A~nowhere dense space of linear superpositions of functions of several variables},
     journal = {Izvestiya. Mathematics },
     pages = {807--837},
     publisher = {mathdoc},
     volume = {6},
     number = {4},
     year = {1972},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1972_6_4_a7/}
}
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B. L. Fridman. A~nowhere dense space of linear superpositions of functions of several variables. Izvestiya. Mathematics , Tome 6 (1972) no. 4, pp. 807-837. http://geodesic.mathdoc.fr/item/IM2_1972_6_4_a7/