On orthogonally additive injections and surjections

Let $E$ be a real inner product space of dimension at least 2 and $V$ a linear topological Hausdorff space. If $\mathrm{card}E\le \mathrm{card}V$, then the set of all orthogonally additive injections mapping $E$ into $V$ is dense in the space of all orthogonally additive functions from $E$ into $V$ with the Tychonoff topology. If $\mathrm{card}V\le \mathrm{card}E$, then the set of all orthogonally additive surjections mapping $E$ into $V$ is dense in the space of all orthogonally additive functions from $E$ into $V$ with the Tychonoff topology.