Bounded linear operators
General · Mathematics
Study notes
Q: Find the norm of T: ℝ²→ℝ², T(x,y) = (2x, 3y) (Euclidean). Is it bounded? ‖T(x,y)‖² = 4x²+9y² ≤ 9(x²+y²) = 9‖(x,y)‖². ‖T‖ ≤ 3. Achieved at (0,1): ‖T(0,1)‖ = 3: ‖T‖ = 3! Bounded ✓ (all finite-dim linear: bounded!). (Operator norm: max stretch!)