Integral Operator is Linear/Corollary 2

Theorem

Let $T$ be an integral operator.

Let $f$ be an integrable real function on a domain appropriate to $T$.

Then:

$\forall \alpha \in \R: \map T {\alpha f} = \alpha \map T f$

Proof

From Integral Operator is Linear:

$\forall \alpha, \beta \in \R: \map T {\alpha f + \beta g} = \alpha \map T f + \beta \map T g$

The result follows by setting $\beta = 0$.

$\blacksquare$

Sources

• 1968: Peter D. Robinson: Fourier and Laplace Transforms ... (previous) ... (next): $\S 1.1$. The Idea of an Integral Transform