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I want to prove that a linear functional $T$ on a normed space $X$ is bounded if and only if $T^{-1}(\{0\})$ is closed. The implication "$\Rightarrow$" is easy. Boundedness of a linear functional is equivalent to beeing continuous. Singletons in normed spaces (which are metric spaces) are closed....

In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X → Y {\displaystyle L:X\to Y} between topological vector spaces (TVSs) X {\displaystyle X} and Y {\displaystyle Y} that maps bounded subsets of X {\displaystyle X} to bounded subsets of Y . {\displaystyle Y.} If X...