Let $B$ be a the set of all real valued functions bounded by $M$. Let $Y$ be a linear subspace of non-negative functions. Define $l$ to be a positive linear functional on $Y$ and let $$p(x) = \inf_{x \leq y} l(y) \quad \forall x\in B, y \in Y$$ We'd like to show $p$ is subadditive.
The proof I'm looking at looks proceeds as follows:
Consider $x_1$ and $x_2$ in $B$ and $y_1$ and $y_2$ in $Y$ such that $x_1 \leq y_1$ and $x_2 \leq y_2$. Adding the two we obtain $x_1 + x_2 \leq y_1 + y_2$ so: $$p(x_1 + x_2) = \inf_{x_1 + x_2 \leq y}l(y) \\ \leq\inf_{x_1 \leq y_1 \\x_2 \leq y_2} l(y) \quad (*)\\ = \inf_{x_1 \leq y_1} l(y_1) + \inf_{x_2 \leq y_2} l(y_2) \quad (*) \\ p(x_1) + p(x_2)$$
I didn't understand how properties of infs were used in the two starred steps. In particular:
- Why inequality between first and second line? Its not obvious to me that the set $\{x_1+x_2 \leq y\} \subset \{x_1 \leq y_1 \cup x_2 \leq y_2\}$
- What's going on with the equality from the second to third line?
Thanks!