Is there always an extension of an operator $T:U\rightarrow W$, defined on (not necessarily closed) subspaces of the infinitedimensional Hilbert spaces $H\supseteq U,L\supseteq W$, to the operator $T':cl(U)\rightarrow cl(W),$ where $cl$ denotes the closure.
Is this extension unique if we require uniform continuity ?
What about the case where $U$ is finite-dimensional ?