# Non-linear projections on subspaces of uniform convex Banach spaces

Reading some parts of Benyamini-Lindenstrauss "Geometric non-linear functional analysis", I got curious about the fact that, in uniformly convex Banach spaces, there exists nonlinear projection on closed convex subsets (so in particular on closed subspaces) [Theorem 2.8 in the afore-mentioned book].

So if $Y, Y' \subset \ell^p(\mathbb{N})$ (mostly curious about the case $p>2$) an element $x$ admits the following five projections $$P_Y x \; , \; P_{Y'} x\; , \; P_{\overline{Y+Y'}}x \; , \; P_Y P_{\overline{Y+Y'}}x \; , \; \text{ and } \; P_{Y'} P_{\overline{Y+Y'}}x.$$ Is there any relation between the norm of these elements? For example, if the norm of $P_{Y'} x$ is small, may one conclude something about the norm of the others? (I'm thinking of something along the line "$P_{\overline{Y+Y'}}x - P_Y x$ is small")

May some sort of orthogonality be used? For example, assuming that, if $j(x) \in \ell^{p'}$ is the linear functionnal obtained by derivating the norm (at $x$), then there exists some small $\epsilon>0$ such that $\forall y' \in Y'$, $\langle j(x) | y' \rangle < \epsilon$. (Or some similar "almost orthogonality" of $Y$ and $Y'$?)

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Of course there are non linear continuous projections onto closed subspaces, but not uniformly continuous ones. In his 1964 paper Lindenstrauss proved that if $Y$ is reflexive and $Y$ is a subspace which is the range of a uniformly continuous non linear projection, there $Y$ is complemented in $X$. –  Bill Johnson Mar 30 '13 at 20:59