# What does the notation $x \in \mathbb{R}^V$ mean where V is a set?

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 ?

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Just to nitpick, why is the union disjoint here? – Alexei Averchenko Jun 8 '11 at 14:44
@Alexei I am sorry i don't understand what you are saying – AnkurVijay Jun 8 '11 at 15:25
Oh, I get it now, silly me :) – Alexei Averchenko Jun 8 '11 at 19:47

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.

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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 8 '11 at 7:11
@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 8 '11 at 7:30
@AnkurVijay an n-tuple is simply a function from $\{1,\ldots,n\}$ (or $n$ as an ordinal) to $\mathbb{R}$. – Alexei Averchenko Jun 8 '11 at 7:42
@Alexei i still dont understand how a single value is being assigned to an n tuple. – AnkurVijay Jun 8 '11 at 7:44
$4$-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 8 '11 at 13:19

It refers to functions that go from Y to X.

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