Inclusion of $\mathbb{L}^p$ spaces, reloaded

Question

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 $p>1$...

Answer 1

An infinite-dimensional example can be obtained as follows:

Let $(Y_n)_{n=1}^\infty \subset L^2[0,1]$ be a sequence of independent standard Gaussians, that is random variables with density $\frac{1}{\sqrt{2\pi}}e^{-t^2/2}$.

Let $X$ be the closed linear span of $(Y_n)_{n=1}^\infty$ in $L^2[0,1]$. I claim that $X \subset L^p$ for all $1 \leq p \lt \infty$.

Consider a finite linear combination $S_N = \sum_{n=1}^N a_n Y_n$. Then $S_N$ is a normal random variable, has mean zero and variance $\sigma^2 = E(|S_N|^2) = \sum_{n=1}^N |a_n|^2$, so $\frac{1}{\sigma} S_N$ is a standard Gaussian, too.

This shows in particular that the space $X$ is isometrically isomorphic to $\ell^2$.

Moreover, we can compute for the $L^p$-norm of $S_N$ as above that $$ \begin{align*} E(|S_N|^p) & = \sigma^p \frac{2}{\sqrt{2\pi}} \int_{0}^\infty t^p e^{-t^2/2}\,dt = \sigma^p \frac{2}{\sqrt{2\pi}} 2^{(p-1)/2} \int_{0}^\infty s^{(p-1)/2} e^{-s}\,ds \\ &= \sigma^p \sqrt{\frac{2^p}{\pi}} \Gamma\left(\frac{p+1}{2}\right), \end{align*} $$ so $\|S_N\|_p = C_p \cdot \|S_N\|_2$ for all $1 \leq p \lt \infty$.

This shows that $X$ is a closed subspace of all spaces $L^p[0,1]$, and up to a constant factor depending only on $p$, its norm is the same as the $L^2$-norm.

Answer 2

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 $1\leq p < \infty$).