In this paper, we have proved that for each cardinal number λ such that λ=λℵ0 a metric space of weight λ admits a bijective continuous mapping onto a Banach space of weight λ. Then, we get that every metric space of weight continuum admits a bijective continuous mapping onto the Hilbert cube. This resolves the famous Banach's Problem (when does a metric (possibly Banach) space X admit a bijective continuous mapping onto a compact metric space?) in the class of metric spaces of weight continuum. Also we get that every metric space of weight λ=λℵ0 admits a bijective continuous mapping onto a Hausdorff compact space. This resolves the Alexandroff Problem (when does a Hausdorff space X admit a bijective continuous mapping onto a Hausdorff compact space?) in the class of metric spaces of weight λ=λℵ0. © 2023 Elsevier B.V.