# Answer to Question #6344 in Functional Analysis for junaid

Question #6344

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

Expert's answer

Not every linear operator between normed spaces is bounded. Let

Define the operator

*X*be the space of all trigonometric polynomials defined on [−π, π], with the normDefine 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.
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