Subspaces of $L^{2}$

Question

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

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

This problem is related to the Erdős–Shapiro–Shields paper [ESS]. From this paper it follows that the answer is negative if $\left\lvert f\right\rvert^{\left\lvert f\right\rvert}$ is replaced by $\left\lvert f\right\rvert^{\left\lvert f\right\rvert^{2}}$.

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 $h :=\left\lvert g\right\rvert^{\left\lvert g\right\rvert}$ belongs to $L^{2}(0,1)$.

Next, it is easy to see that $\left\lVert f\right\rVert _{p}^{p}\leq1+\left\lVert h\right\rVert _{2}^{2}<+\infty$.

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 $j:(F, \lVert{.}\rVert_{2})\rightarrow L^{p}(0,1)$, 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] Erdős, Shapiro, and Shields - Large and small subspaces of Hilbert space.

Answer 1

The classical lacunary series example allows you to integrate anything that is $e^{O(|f|^2)}$, so it works for your question. It seems that for reasonable (say, positive, increasing, and convex) functions $\Phi$, the complete answer is the following: A closed infinite-dimensional subspace $F$ such that $\Phi(|f|)$ is integrable for all $f\in F$ exists if and only if $\log\Phi(x)=O(x^2)$. The reason is that you can always create a random function in the unit ball that is pointwise almost Gaussian by taking a random linear combination of large number $n$ of the elements in the orthonormal basis of $F$ with random coefficients of size $n^{-1/2}$.