Answer to Question #6344 in Functional Analysis for junaid

Question

X is a Normed linear space. T is a function from X to X. Then T is a bounded linear transformation

Answer 1

Not every linear operator between normed spaces is bounded. Let X be the space of all trigonometric polynomials defined on [−π, π], with the norm

Define the operator L:X→X which acts by taking the derivative, so it maps a polynomial P to its derivative P′. Then, for
v = e^inx with n=1, 2, ...., we have while as n→∞, so this operator is not bounded.