Linear Transformation between Normed Vector Spaces is Bounded iff Bounded as Linear Transformation between Topological Vector Spaces

Theorem
Let $\GF \in \set {\R, \C}$.

Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be normed vector spaces over $\GF$.

Let $T : X \to Y$ be linear transformations.

Then $T$ is bounded as a linear transformation between normed vector spaces it is bounded as a linear transformation between topological vector spaces.

Sufficient Condition
Suppose that $T$ is bounded as a linear transformation between topological vector spaces

Then:


 * for each von Neumann-bounded subset $E$ of $X$, $T \sqbrk E$ is von Neumann-bounded.

From Characterization of von Neumann-Boundedness in Normed Vector Space, this is equivalent to:


 * if $E \subseteq X$ is such that $\norm x_X < M$ for each $x \in E$ and some $M > 0$, then there exists $M' > 0$ such that $\norm {T x}_Y < M'$ for each $x \in E$.

In particular:


 * there exists $M' > 0$ such that $\norm {T x}_Y < M'$ for each $x \in X$ with $\norm x_X < 1$.

Then for $x \ne \mathbf 0_X$, we have:


 * $\ds \norm {\map T {\frac x {2 \norm x_X} } }_Y < M'$

so that:


 * $\ds \norm {T x}_Y < 2 M' \norm x_X$

for each $x \ne \mathbf 0_X$.

Hence $\norm {T x}_Y \le 2 M' \norm x_X$ for all $x \in X$.

So $T$ is bounded as a linear transformation between normed vector spaces.

Necessary Condition
Suppose that bounded as a linear transformation between normed vector spaces.

Again from Characterization of von Neumann-Boundedness in Normed Vector Space, we need to show that:


 * if $E \subseteq X$ is such that $\norm x_X < M$ for each $x \in E$ and some $M > 0$, then there exists $M' > 0$ such that $\norm {T x}_Y < M'$ for each $x \in E$.

Let $E \subseteq X$ be such that $\norm x_X < M$ for each $x \in E$.

From the definition of a bounded linear transformation between normed vector spaces, there exists $K > 0$ such that:


 * $\norm {T x}_Y \le K \norm x_X$

for each $x \in X$.

Then for $x \in E$ we have $\norm {T x}_Y < M' K$.

So $T$ is bounded as a linear transformation between topological vector spaces.