Let \(T\colon X\to Y\) be a linear map between normed spaces . Then the following statements are equivalent:

  1. \(T\) is continuous .
  2. \(T\) is continuous in \(0\).
  3. The is a constant \(M>0\) such that \(\lVert Tx\rVert\le M\lVert x\rVert\) for every \(x\in X\).
  4. \(T\) is uniform continuous.
Remarks
  • Due to the third statement, \(T\) maps the unit ball in \(X\) onto a bounded set in \(Y\). This motivates the name bounded operator .