Arg Sup vs Arg Max

[spl]

Let X be the space of functions from R -> R

Comments

[improvedhuman]

I have never seen "arg sup". Curious about where you are encountering it.

If it really means "the value of x at which f(x) attains its supremum," your reasoning looks correct.

[lifeofbai]

Unless I'm misreading this...

I don't think max(f(x)) is necessarily the same as sup(f(x)). For example, consider f(x) = (2x^2-11)/(x^2+9).

sup(f(x)) = 2, but there is no maximal element (there is no value of x for which f(x) = 2).

- Ray

[spl]

They key is the "arg". In that case, the arg sup would be undefined. I think...

[lifeofbai]

Yeah, you're correct. I was wrong. :)

[lifeofbai]

I think you are right. If arg max is defined, then it would also be the supremum.

- Ray