# Concept derivation guides

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Concept guide



## [Bounded linear operator: exact induced operator norms of a matrix under ℓ^∞ and ℓ¹](/en/derivations/derivation-mko-bounded-linear-operator-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-bounded-linear-operator)

Let A be the integer matrix with first row (1,−2) and second row (3,4), and T(x)=Ax:ℝ²→ℝ². (1) With the ℓ^∞ norm at both ends, prove that ‖Ax‖_∞≤7‖x‖_∞ for all x and that no smaller constant works, so ‖T‖_{∞→∞}=7 (the maximum absolute row sum). (2) With the ℓ¹ norm at both ends, prove ‖T‖_{1→1}=6 (the maximum absolute column sum). (3) Explain three traps: non-optimal constants, finitely many inputs, and unbounded range.

Bounded linear operator

derivation-mko-bounded-linear-operator-01





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