What does notation $\inf_{k,l}$ mean for indices $k,l$?
Does it mean that one picks $\inf k$ and $\inf l$ or that one picks some kind of "inf of both $k$ and $l$" (whatever that means)?
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It probably means you are taking the infimum over $k$ and $l$, for something that's a function of $k$ and $l$. – Minus One-Twelfth May 20 '19 at 10:34
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@MinusOne-Twelfth $k,l$ are indices as I said. So one can for example assume that $k \in {1,2,...,n},l \in {1,2,...,n}$. But is the $\inf$ then $1$ and $1$ or $(1,1)$? And if it's $(1,1)$, then this being $\inf$ may not be trivial, because what's the measure? – mavavilj May 20 '19 at 10:41
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In that case, it would be the minimum value of the function for $k,l\in{1,\ldots,n}$. You must have a function or set or something that you are taking the inf of. For example $\inf\limits_{k,l}a_{k,l}$, where $A=(a_{k,l})$ is some given matrix. – Minus One-Twelfth May 20 '19 at 10:42
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@MinusOne-Twelfth Ah you mean that it asks to "select $k,l$ such that the expression after $\inf$ attains its $\inf$"? – mavavilj May 20 '19 at 10:43
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Almost, but $\inf\limits_{k,l}a_{k,l}$ does not refer to picking the $k,l$, rather it refers to what this infimal value of $a_{k,l}$ is. For example, if $A=(a_{k,l})$ is a matrix whose least entry is $-2$, then $\inf\limits_{k,l}a_{k,l}=-2$. – Minus One-Twelfth May 20 '19 at 10:45
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Unless the context clearly explains a different convention I would expect $$ \inf_{k,l} f(k,l) $$ to mean the infimum of the set $$ \{ f(k,l) \mid (k,l)\in K\times L \}$$ which is hopefully a subset of $\mathbb R$ or some other ordered set where you can speak of infima.
You'll need to deduce from the context what the index sets $K$ and $L$ are.
Note that if there are infinitely many possible $k$s or $l$s, the set above can be infinite, and in that case "infimum" may not be the same as "minimum"
hmakholm left over Monica
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