# Banach lattice subspace of $C([0,1])$ not a sublattice

This is probably easy, but I did not see it in standard texts. Describe a closed subspace $V$ of $C([0,1])$ such that $V$ is a Banach lattice (in the pointwise ordering), but $V$ is not a sublattice of $C([0,1])$. Note that, by virtue of the formula $2(f\vee g) = f+g+|f-g|$, such a $V$ must contain an element $h$ such that $|h| \notin V$.

Maybe there is a simple example using some other Banach lattice $L$ in place of $C([0,1])$, and exhibiting a closed subspace $V$ of $L$ which is a Banach lattice in the ordering inherited from $L$, but not a sublattice.

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Yeah, it's easy, take $V$ to be the set of linear functions $ax + b$. Any $f$ and $g$ in $V$ have a least upper bound which is the line from $\max(f(0),g(0))$ to $\max(f(1),g(1))$. The sup norm of $f \vee -f$ equals the sup norm of $f$, so it's a Banach lattice.

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Thanks. Nice example. –  Fred Dashiell Jan 16 at 14:29
A vector subspace $X$ of a vector lattice $E$ is a vector lattice-subspace if $X$ equipped with the order induced by $E$ is a vector lattice. The space is a vector sublattice if $x\vee y$ and $x\wedge y$ are in $X$ for every $x,y\in X$.
A sublattice is a lattice-subspace but the converse need not be true. Every two dimensional subspace $X\subseteq E$ such that $X_+=X\cap E_+$ is also two dimensional is a lattice-subspace. To see this notice that in the two dimensional case $(X_+ + x) \cap (X_+ + y)$ is always a translation of the cone $X_+ + z$ and $z$ is the supremum of $\{x,y\}$. Furthermore, in $E=L_p$ (and $C([0,1])$) if such a space $X$ contains the constant function 1 and a strictly monotone function it is not a sublattice. To see this notice that the smallest lattice $Y$ containing $X$ will not be two dimensional because in $L_p$ the closure of $Y$ is the $p$-integrable functions that are measurable in the $\sigma$-algebra generated by the functions in $X$.
Theorem: Let $X$ be a subspace of a vector lattice $E$ and let $Y$ be the smallest vector sublattice of $E$ containing $X$. The vector space $X$ is a lattice-subspace if and only if there is a positive linear projection of $Y$ onto $X$.