TITLE: Inclusion of $\mathbb{L}^p$ spaces, reloaded
QUESTION [6 upvotes]: I have a follow-up from this question. It was proved that, if $X$ is a linear subspace of $\mathbb{L}^1 (\mathbb{R})$ such that:

$X$ is closed in $\mathbb{L}^1 (\mathbb{R})$;
$X \subset \bigcup_{p > 1} \mathbb{L}^p (\mathbb{R})$,

then $X \subset \mathbb{L}^p (\mathbb{R})$ for some $p>1$.
I was wondering whether one could find a subspace $X$ satisfying these hypotheses and which is infinite-dimensional. It turns out this is possible. If one chooses a bump function, and considers the closure for the $\mathbb{L}^1 (\mathbb{R})$ norm of the space generated by the translates by integers of this bump function, one can emulate the $\ell^1$ space. The resulting $X$ will be closed, and included in $\mathbb{L}^p (\mathbb{R})$ for all $p>0$. To avoid this phenomenon, I'll restrict myself to smaller spaces.

Is there a linear, closed, infinite-dimensional subspace $X$ of $\mathbb{L}^1 ([0,1])$ which is included in $\mathbb{L}^p ([0,1])$ for some $p>1$?

The problem is that any obvious choice of countable basis will very easily generate all of $\mathbb{L}^1 ([0,1])$ (polynomials, trigonometric polynomials...), or $\mathbb{L}^1 (A)$ for some $A \subset [0,1]$, or at least one function which is in $\mathbb{L}^1$ but not in $\mathbb{L}^p$ for $p>1$...

REPLY [3 votes]: There exists such space. Let $\{R_i\}$ be the set of Rademacher functions on $[0,1]$. Specifically, let 
$$R_k(x) = sign(\sin(\pi k x)).$$
For every $\alpha = (\alpha_1, \dots, \alpha_k, \dots) \in \ell_2$, let $f_{\alpha}(x) = \sum \alpha_k R_k(x)$. Note that by the Khintchine inequality, for every $p\in [1,\infty)$, 
$$A_p \|\alpha\|_2 \leq \|f_{\alpha}\|_p \leq B_p \|\alpha\|_2$$ 
(where $A_p$ and $B_p$ depend only on $p$). In particular, $\|f_{\alpha}\|_1 < \infty$ for all $\alpha\in \ell_2$. Therefore, $f_{\alpha} \in L_1[0,1]$ for every $\alpha \in \ell_2$. Consider $V = \{f_{\alpha}: \alpha\in \ell_2\}$. 
Clearly, $V$ is a linear subspace of $L_1[0,1]$. Moreover, $V$ is a closed subspace. Indeed if a sequence $f_{\alpha^{(1)}},f_{\alpha^{(2)}}, f_{\alpha^{(3)}}, \dots$ converges to $g\in L_1[0,1]$, then $f_{\alpha^{(k)}}$ is a Cauchy sequence in $L_1[0,1]$. Therefore, by the Khinchine Inequality, $\alpha^{(k)}$ is a Cauchy sequence in $\ell_2$. Hence it converges to some $\alpha^* \in \ell_2$. Then $f_{\alpha^{(k)}} \to f_{\alpha^*}$, and $g = f_{\alpha^*}$ (a.e.).
The Khintchine inequality implies that $V$ is a subspace of every $L_p[0,1]$ (where $1\leq p < \infty$).