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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 X be the space of all trigonometric polynomials defined on [−π, π], with the norm
http://upload.wikimedia.org/wikipedia/en/math/c/5/3/c5399072cdb8a4ff4fd2cf29ce7c7cc8.png
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 = einx with n=1, 2, ...., we have http://upload.wikimedia.org/wikipedia/en/math/5/b/c/5bc2e9c2e6f85669f73fa651473217fb.png while http://upload.wikimedia.org/wikipedia/en/math/a/a/7/aa795880ad03f0bd5c151176470e09f8.png as n→∞, so this operator is not bounded.

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