I have a problem where I need to work with functions that are square-integrable, bounded and continuous, i.e. the space
$ L^2 \supset X = \left\{ f \in L^2 \mid f \text{ bounded, continuous}\right\} $
equipped with the $L^2$ norm.
I have a hard time understanding if this is a subspace of $L^2$ because I do not know how to show whether or not $X$ is complete under the $L^2$ norm.
Any text regarding this would be appreciated.