Let $T$ be an element of $B(X,X)$ (the bounded linear operators from X to itself), and let $W$ be a subset of $X$. Show that $T(\overline{W}) \subset \overline{T(W)]}$. Furthermore, if $T$ has bounded inverse, then $T(\overline{W}) = \overline{[T(W)]}$.
Invertible bounded linear operators and closure
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functional-analysis
banach-spaces
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0Try \bar{W} ($\bar{W}$) or \overline{W} ($\overline{W}$). – 2011-05-10
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1Use `\overline{...}`. Concerning your question. You don't say what $X$ is but I guess it's a Banach space. Note that this has *nothing* to do with $X$ being a Banach space and linearity of $T$ (except for the trivial fact bounded = continuous). It's just an equivalent reformulation of continuity of $T$ (and of $T^{-1}$) between topological spaces. – 2011-05-10