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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...
