Bounded modular functionals and operators on Hilbert C*-modules are regular

We prove that for any C*-algebra $A$ and Hilbert $A$-modules $M\subseteq N$ with $M^\perp=\{0\}$, every bounded $A$-linear map $N\to A$ (or $N\to N)$ vanishing on $M$ is the zero map. This verifies the conjectures of the first author and settles the regularity problem for bounded modular functionals and operators on Hilbert C*-modules. As a consequence, kernels of bounded C*-linear operators on Hilbert C*-modules are shown to be biorthogonally complemented, which gives a correct proof of Lemma 2.4 in ``On Hahn-Banach type theorems for Hilbert C*-modules'', Internat. J. Math. 13(2002), 1--19, in full generality.