The infinite-dimensional Hilbert sphere ${\mathcal{S}}^{\mathrm{\infty }}$ has been widely employed to model density functions and shapes, extending the finite-dimensional counterpart. We consider the Fréchet mean as an intrinsic summary of the central tendency of data lying on ${\mathcal{S}}^{\mathrm{\infty }}$. For sound statistical inference, we derive properties of the Fréchet mean on ${\mathcal{S}}^{\mathrm{\infty }}$ by establishing its existence and uniqueness as well as a root-n central limit theorem (CLT) for the sample version, overcoming obstructions from infinite-dimensionality and lack of compactness on ${\mathcal{S}}^{\mathrm{\infty }}$. Intrinsic CLTs for the estimated tangent vectors and covariance operator are also obtained. Asymptotic and bootstrap hypothesis tests for the Fréchet mean based on projection and norm are then proposed and are shown to be consistent. The proposed two-sample tests are applied to make inference for daily taxi demand patterns over Manhattan, modeled as densities, of which the square root densities are analyzed on the Hilbert sphere. Numerical properties of the proposed hypothesis tests which utilize the spherical geometry are studied in the real data application and simulations, where we demonstrate that the tests based on the intrinsic geometry compare favorably to those based on an extrinsic or flat geometry.