# Example to linear but not continuous

We know that when $(X,\|\cdot\|_X)$ is finite dimensional normed space and $(Y,\|\cdot\|_Y)$ is arbitrary dimensional normed space if $T:X \to Y$ is linear then it is continuous (or bounded)

But I cannot imagine example for when $(X,\|\cdot\|_X)$ and $(Y,\|\cdot\|_Y)$ are arbitrary dimensional normed spaces $T:X \to Y$ is linear and not bounded or continuous.

Could someone give any simple example please?

Thanks

• Possible duplicate of Does existence of a non-continuous linear functional depend on Axiom of Choice? Jun 11 '18 at 16:31
• @user73985 : how is the question here related to the axiome of choice? Jun 11 '18 at 16:48
• @Watson that question is strictly stronger than this one; its accepted answer includes a (simple) answer to this one Jun 11 '18 at 16:50

## 1 Answer

The differentiation operator is noncontinuous (not bounded) on the space $\Bbb R[x]$ of all polynomials with $\sup$ norm over $[-1,1]$.

• Can I use a sequence like $f_n(x)=x^n$ to show unboundedness? Jun 11 '18 at 16:56
• Yes. Or, alternatively, $\frac{x^n}n\, \to0$ while its image $x^{n-1}$ doesn't tend to $0$. (Taking e.g. the sup norm over $[-1,1]$.) Jun 11 '18 at 17:05