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.
2026-04-02 11:57:39.1775131059
Differentiation Operator is not a bounded operator for polynomials
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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.