fa.functional analysis - Subspaces of $L^{2}$

[In what follows $0^{0}$= 1 by convention.]

Is there some closed infinite dimensional linear subspace $F$ of $L^{2}(0,1)$
such that $left|fright|^{left|fright|}$ belongs to $L^{2}(0,1)$
for all $f$ in $F$ ?

This problem is related to the Erdos - Shapiro - Shields paper [ESS].
From this paper it follows that the answer is negative if $left|fright|^{left|fright|}$ is replaced by $left|fright|^{left|fright|^{2}}$.

Some thoughts. Suppose that such an $F$ exists, and take some $p > 2$.
Let $f$ be in $F$.

Then clearly $g:=(p/2)cdot f$ is in $F$, too, hence $h :=left|gright|^{left|gright|}$
belongs to $L^{2}(0,1)$.

Next, it is easy to see that $leftVert frightVert _{p}^{p}leq1+leftVert hrightVert _{2}^{2}<+infty$.

Therefore, $F$ is contained in $L^{p}(0,1)$ as a linear subspace
(i.e., algebraically).

Now, applying the Closed Graph Theorem to the natural linear embedding
$j:(F, ||.||_{2})rightarrow L^{p}(0,1)$, it follows that $j$ is
continuous. Consequently, the Hilbertian 2-norm and the $p$-norm
are equivalent on $F$. Moreover, it follows that $F$ is complete
w.r.t. the $p$-norm, and, in turn, it is a closed subspace of $L^{p}(0,1)$.

And this is true for all $p > 2$.

[ESS] http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.mmj/1028999306