Geodesic
The norm in $C$ of orthogonal projections onto subspaces of polygonal functions
Matematičeskie zametki, Tome 21 (1977) no. 4, pp. 495-502

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Let $P_\pi$ be an orthogonal projection (in the sense of $L_2$) onto the subspace of polygonal functions over a certain partition $\pi$ of the segment $[0,1]$. Z. Ciesielski has established the following estimate for the norm of this operators, as acting from $C$ into $C$, valid for an arbitrary partition: $\|P_\pi\|_{C\to C}\le3$. In this note it is proved that this estimate is final; more precisely, it is shown that $\sup\limits_\pi\|P_\pi\|_{C\to C}=3$. 
@article{MZM_1977_21_4_a5,
     author = {P. Oswald},
     title = {The norm in $C$ of orthogonal projections onto subspaces of polygonal functions},
     journal = {Matemati\v{c}eskie zametki},
     pages = {495--502},
     publisher = {mathdoc},
     volume = {21},
     number = {4},
     year = {1977},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1977_21_4_a5/}
}
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P. Oswald. The norm in $C$ of orthogonal projections onto subspaces of polygonal functions. Matematičeskie zametki, Tome 21 (1977) no. 4, pp. 495-502. http://geodesic.mathdoc.fr/item/MZM_1977_21_4_a5/