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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

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

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  • $\begingroup$ Can I use a sequence like $f_n(x)=x^n$ to show unboundedness? $\endgroup$ – user519955 Jun 11 '18 at 16:56
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    $\begingroup$ 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 $[-1,1]$.) $\endgroup$ – Berci Jun 11 '18 at 17:05

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