Monotone operator on $L^2(0,\infty)$

Question

I trying to prove the following assestment

Every linear monotone operator on $L^2 (0, \infty)$ is bounded

Any ideas?

Thank you

Answer 1

Hint: Let $T: L^2(0,\infty)\to \mathbb R$ be a monotone, linear operator. Think about the following: If $f_1, f_2, \dots$ is a suitable sequence of non-negative functions in $L^2$, then $$ \sum_{n=1}^\infty T(f_k)\le T\left(\sum_{n=1}^\infty f_n\right).$$ Assuming the operator was not continuous, you can use this to obtain a contradiction (note that $T(f) = \infty$ is not allowed for any $f\in L^2$).

Answer 2

Generally, every monotone (possibly nonlinear, not everwhere defined or even multi-valued) operator is locally bounded in the interior of its domain. This is not so trivial and can be founded in every good book treating monotone operators. See e.g. Zeidler - Nonlinear functional analysis 2B, Prop. 32.33, p. 884.

Hence, because your operator is defined everywhere and linear, it is therefore a bounded operator.