Let $P$ be the set of all polynomials, with complex coefficients, in a variable $t$. For $x$ in $P$ the function $y$ is defined by $y\left( x\right) =\dfrac {d^{2}x} {dt^2}|_{t=1}$ Is $y$ a linear functional? Now I am aware of $y$ being a scalar valued function and the defining property of a linear functional is $y\left( \alpha _{1}x_{1}+\alpha_{2}x_{2}\right) =\alpha _{1}y\left( x_{1}\right) +\alpha _{2}y\left( x_{2}\right)$ It is the actual definition of $y$ which I am confused about.
Any help would be much appreciated.