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In the context of submodular functions, I encountered the following statement :

For a vector $x \in \mathbb{R}^V$ and a subset $Y \subseteq V$ we define the expression $x(Y)$ as $\sum_{u \in Y}x(u)$.

$V$ is a set.

What does this statement mean ?

AnkurVijay
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2 Answers2

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For sets $X$ and $Y$ the notation $X^Y$ means the following:

$$ X^Y = \{f:Y \to X \mbox{ function}\} $$

if $X$ is a field, then $X^Y$ can be given a structure of vector space over $X$ with the obvious point-wise operations.

Giacomo d'Antonio
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    To elaborate further, one can see that this is in some way consistent with the notation $\mathbb{R}^n$ which can be viewed as the set of functions from a set of $n$ elements to $\mathbb{R}.$ – Kopper Jun 08 '11 at 07:11
  • @Jay how can the notation $x \in \mathbb{R}^n$ be interpreted as you suggest ? I only know of the interpretation that x is simply a vector of n elements each one of which belongs to $\mathbb{R}$. Please elaborate a little more. – AnkurVijay Jun 08 '11 at 07:30
  • @AnkurVijay an n-tuple is simply a function from ${1,\ldots,n}$ (or $n$ as an ordinal) to $\mathbb{R}$. – Aleksei Averchenko Jun 08 '11 at 07:42
  • @Alexei i still dont understand how a single value is being assigned to an n tuple. – AnkurVijay Jun 08 '11 at 07:44
  • @AnkurVijay to $f:{1, \dots, n} \to \mathbb R$ assign $(f(1), \dots, f(n))$ – Giacomo d'Antonio Jun 08 '11 at 08:44
  • @Giacomo sorry for being a little dumb here, but could you explain to me with an example ? $x \in \mathbb{R}^n$ to me means an n tuple each element of which is in $\mathbb{R}$. For example x = (5,3,7.5, -10) is a 4-tuple. Now how do i give this a single value belonging to $\mathbb{R}$. What is $f$ in your comment above ? – AnkurVijay Jun 08 '11 at 11:23
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    $4$-tuple $x=(5,3,7.5,-10)$ corresponds to function $f$, with domain ${1,2,3,4}$ where $f(1) = 5, f(2)=3, f(3)=7.5, f(4)=-10$ – GEdgar Jun 08 '11 at 13:19
  • @GEdgar thankyou very much, now i understand what is being suggested – AnkurVijay Jun 08 '11 at 15:26
  • @AnkurVijay: perhaps this old answer of mine is helpful: http://math.stackexchange.com/a/51062/2614 – Bruno Stonek Feb 02 '12 at 02:03
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It refers to functions that go from Y to X.

davidlowryduda
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