When working with operators between normed spaces in machine learning, you often need to ensure that your transformations or mappings behave predictably under small changes. Two key concepts that describe this behavior are continuity and boundedness of operators.

A linear operator $T$ between normed spaces $X$ and $Y$ is said to be continuous if, whenever a sequence $(x_n)$ in $X$ converges to some $x$ in $X$, the sequence $(T(x_n))$ converges to $T(x)$ in $Y$. In practical terms, this means that small changes in input lead to small changes in output, which is essential for stability in learning algorithms.

On the other hand, a linear operator is bounded if there exists a constant $C \geq 0$ such that for all $x$ in $X$, the norm of $T(x)$ is less than or equal to $C$ times the norm of $x$. Formally,
$||T(x)||_Y \leq C * ||x||_X$ for all $x$ in $X$.
This condition guarantees that the operator does not amplify the input uncontrollably, another crucial property for algorithms that must generalize well from data.

A fundamental result in functional analysis states that for linear operators between normed spaces, continuity and boundedness are equivalent. This equivalence is both elegant and practically important.

Theorem:
Let $T: X \rightarrow Y$ be a linear operator between normed spaces. Then $T$ is continuous if and only if $T$ is bounded.

Proof Sketch:
Suppose $T$ is bounded. Then, for any sequence $(x_n)$ converging to $x$ in $X$,
$||T(x_n) - T(x)|| = ||T(x_n - x)|| \leq C * ||x_n - x||$.
Since $||x_n - x|| \rightarrow 0$, it follows that $||T(x_n) - T(x)|| \rightarrow 0$, so $T$ is continuous.

Conversely, if $T$ is continuous at $0$, then there exists some $\delta > 0$ such that $||x|| < \delta$ implies $||T(x)|| < 1$. Using linearity and scaling, you can show that this implies the existence of a constant $C$ such that $||T(x)|| \leq C * ||x||$ for all $x$. Thus, $T$ is bounded.

This result allows you to use either property depending on which is easier to verify or more natural in your application.