Landau's converse to H\" older's inequality

By H" older’s inequality, if $\mathbf{x}\in {\mathrm{\ell }}^p$, then $\mathbf{x}\mathbf{y}\in {\mathrm{\ell }}^1$ for all $\mathbf{y}\in {\mathrm{\ell }}^q$. Landau proved the converse result: If $\mathbf{x}\mathbf{y}\in {\mathrm{\ell }}^1$ for all $\mathbf{y}\in {\mathrm{\ell }}^q$, then $\mathbf{x}\in {\mathrm{\ell }}^p$. This paper proves Landau’s theorem and considers a related problem for Hilbert’s inequality.