TITLE: Riesz space of functions from any set $X$ to $\mathbb{R}$
QUESTION [1 upvotes]: We collect certain functions from $f:X\to\mathbb{R}$ to $\mathscr{T}$ so that it is a vector space over $\mathbb{R}$ under the usual function addition, scalar product. Now we define an partial order on it: $f\leq g$ iff for all points $x\in X, f(x)\leq f(x)$. This is compatible with the vector space structure. Now we claim merely  $\mathscr{T}$ as a Riesz space without giving exact formula of infimum or supremum. Can I show the formula is $f\wedge g=\min\{f,g\}$? The thing is if we just first collect functions but we might not be sure that I have collected $\min\{f,g\}$.

REPLY [0 votes]: I'm not sure, but it seems like possibly you're asking whether any vector space of functions from $X$ to $\Bbb R$ is a Riesz space, with the natural partial order. If that's the question the answer is no; for example the space of all polynomials is not a lattice.
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