Isometric embedding of U into a Hilbert space

Determine whether there exists an isometric embedding of the Banach space U of arithmetic functions, equipped with the norm ||f||_U = sup_{n in N} |f(n)| / log(2+n), into a Hilbert space to enable the application of Fourier-analytic and Hilbert space techniques.

Background

Although U is presented as a Banach space that balances Hilbert-like control of Dirichlet coefficients and Banach-type control of logarithmic averages, it is not established whether U can be realized within a Hilbert space without distortion. Such an embedding would unlock a broad toolkit from Hilbert space theory.

The authors explicitly ask whether U admits an isometric embedding into a Hilbert space, framing a structural question about the geometry of U and its compatibility with inner-product methods.