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 =
einx with
n=1, 2, ...., we have
![https://upload.wikimedia.org/wikipedia/en/math/5/b/c/5bc2e9c2e6f85669f73fa651473217fb.png](https://upload.wikimedia.org/wikipedia/en/math/5/b/c/5bc2e9c2e6f85669f73fa651473217fb.png)
while
![https://upload.wikimedia.org/wikipedia/en/math/a/a/7/aa795880ad03f0bd5c151176470e09f8.png](https://upload.wikimedia.org/wikipedia/en/math/a/a/7/aa795880ad03f0bd5c151176470e09f8.png)
as
n→∞, so this operator is not bounded.
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