Does the Sup/Inf Operator Apply in Vector/Matrix Conditions?

[pitaly]

Suppose y is a positive vector. Let p and x be two positive matrices with N rows, where $p_j$ and $x_j$ denotes the j:th row in these matrices, so that j = 1,…,N.

Does the following hold:

\inf_{k=1,...,N} [\sup_{l=1,...,N} [p_k(y-x_k)]] = \inf_{k=1,...,N} [p_k(y-x_k)]

where $p_k(y-x_k)$ is a scalar.

That is, given that I don’t take sup over the sub indices k but over the sub indices l which isn’t part of the expression $p_k(y-x_k)$ will I be able to skip the sup in the expression?

If this is true, then I should also be able to write

\sup_{k=1,...,N} [\inf_{l=1,...,N} [p_l(y-x_l)]] = \inf_{l=1,...,N} [p_l(y-x_l)]

 

[andrewkirk]

Hello pitaly and welcome to physicsforums!

What do you mean by 'positive vector', and what do you mean by 'positive matrix'?

Also, your post will be easier to read if you use two consecutive # marks as opening and closing delimiters for latex when it is part of a sentence (eg your first two latex instances), as opposed to an equation that you wish to display on a separate line, for which you use two consecutive $ signs. The # approach gives what is called 'in-line' latex, which is achieved by a single $ in other latex implementations.

 

[andrewkirk]

Actually, I think the answer to your question is Yes, regardless of what 'positive' means, even if it doesn't impose any restrictions.

That's because $$\inf_{x\in S}[\textrm{expression E, which doesn't involve }x]=\textrm{expression E}$$

as long as $S$ is non-empty. And the same applies to sup.