Uniqueness of a linear operator The wikipedia entry for bounded operators shows that for the space $X$ of trigonometric polynomials on $[-\pi,\pi]$ with norm $$\lVert P\rVert = \int_{-\pi}^{\pi}\lvert P(x)\rvert \operatorname{d}\\!x,$$ the operator $L: X\rightarrow X$ by $P\mapsto P^{\prime}$ is not a bounded operator. It is not clear to me why there couldn't be something like a bounded linear operator $K$ on a larger space $X\subseteq Y$ such that the restriction of $K$ onto $X$ agrees with $L$. Is there an obvious uniqueness theorem I am missing? I suspect that this can be proven by contradiction using the Riesz representation theorem, but I think this might be overkill and I am missing a comparatively trivial argument.

The boundedness depends of the norm. $X\subseteq Y$ must be interpreted not only as set-theoretical inclusion, but that $X$ inherits the norm of $Y$.