Differentiation Operator is not a bounded operator for polynomials If you consider the space of all polynomials on [0,1] (defined as $P_{[0,1]}$ as subspace of $C_{[0,1]}$) then the differentiation operator is not a linear bounded operator on this space. Why is that? this doesn'
t make any sense to me. 

There's no reason to assume that the operator is bounded here. The norm on $C_{[0, 1]}$ measures the size of a function in terms of its values, but the derivative is really about steepness. A function can be arbitrarily steep even though it takes only small values.
To be explicit, the norm of $x^n$ is clearly $1$, while the norm of its derivative is $n$. Draw the picture to see how $x^n$ gets progressively steeper while not getting bigger in value.
