Let {f(adsbygoogle = window.adsbygoogle || []).push({}); _{i}}_{i E I}be a family of real-valued functions R^{n}->R.

Define a function

f(x)

=sup f_{i}(x)

i E I

1) I'm having some trouble understanding what the sup over i E I of a function of x means? The usual "sup" that I've seen is something like

supf(x)

x E S

for some set S.

But they instead have i E I there which confuses me.

2) Is the following true?

sup [c * f_{i}(x)]

i E I

= c *sup [f_{i}(x)]

----i E I

In other words, can we pull out a constant out of the sup? If so, how can we rigorously prove it?

Any help is appreciated!

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# Supremum of function

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