Bounded linear operators

Objective. Calculate operator norms and continuity.

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2. Worked example
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Learn: the key idea

A linear operator T:V→W between normed spaces is bounded if there exists C≥0 with ‖Tv‖≤C‖v‖ for all v, and the smallest such C is the operator norm ‖T‖=sup_{‖v‖=1}‖Tv‖; boundedness for linear operators is equivalent to continuity, unlike for general functions. Bounded operators form a normed vector space themselves under the operator norm, and composition of bounded operators is bounded with ‖ST‖≤‖S‖‖T‖. For example, the operator T:ℝ²→ℝ² given by the matrix [[2,0],[0,3]] has operator norm ‖T‖=3 (with the Euclidean norm) since the largest stretching factor it applies to any unit vector is 3.

Worked example

Find the operator norm of the bounded linear operator T(x,y)=(4x,0) on ℝ² with the Euclidean norm.

1 $‖T(x,y)‖ = ‖(4x,0)‖ = 4|x|$
2 $For unit vectors, x²+y²=1, so |x|\leq1 with equality when y=0$
3 $Maximum of 4|x| over unit vectors is achieved at x=1,y=0, giving value 4$
4 $Operator norm ‖T‖ = 4$

Answer: 4

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Common mistake. Students compute operator norms by testing only basis vectors instead of all unit vectors, missing the true supremum; for a matrix like [[1,1],[0,1]], checking just e₁ and e₂ underestimates the norm, which actually comes from a combined direction.

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