Orlicz space and Hilbert space

Question

Lindenstrauss and Tzafriri state that the spaces

$\ell_M=\left\{x=(x_1,...):\inf\{\rho>0:\sum_{j=1}^{\infty}M(\frac{x_j}{\rho})\leq 1)\}<\infty\right\}$ and

$h_M=\left\{x=(x_1,...):\sum_{j=1}^{\infty}M(\frac{x_j}{\rho})<\infty\text{ for every }\rho>0 \right\}$

are identical and separable if $M$ satisfies the $\Delta_2$-condition. Of course for $M(t)=t^2$ and similar functions this is the well known space $\ell^2$. I tried to think of an Orlicz function such that $\ell_M$ is a Hilbert space other than $\ell^2$ but was not successful so far. Doe´s anybody know an example? Many Thanks in advance.

Answer 1

There is a degree of ambiguity in your question. A Banach space may become Hilbert under an equivalent norm. If you require the Orlicz space in question to be Hilbert in the Minkowski gauge norm (Luxemburg norm) it is shown in this post that $x^2/2$ is the only function conjugate to itself.

For the Hilbert property in a wider sense see Rao, M.M., Notes on characterizing Hilbert space by smoothness and smooth Orlicz spaces, J. Math. Anal. Appl. 37, 228-234 (1972).