Banach lattice

In mathematics, specifically in functional analysis and order theory, a Banach lattice $(X,\|\cdot \|)$ is a Riesz space with a norm $\|\cdot \|$ such that $(X,\|\cdot \|)$ is a Banach space and for all $x,y\in X$ the implication $|x|\leq |y|\Rightarrow \|x\|\leq \|y\|$ holds, where as usual $|x|:=x\vee -x$.

Examples and constructions

• $\mathbb {R}$, together with its absolute value as a norm, is a Banach lattice.
• Let $X$ be a topological space, $Y$ a Banach lattice and ${\mathcal {C}}(X,Y)$ the space of bounded, continuous functions from $X$ to $Y$ with norm $\|f\|_{\infty }:=\sup _{x\in X}\|f(x)\|_{Y}$. ${\mathcal {C}}(X,Y)$ becomes a Banach lattice with the pointwise order $f\leq g:\Leftrightarrow \forall x\in X:f(x)\leq g(x)$.

See also

• Banach space
• Riesz space
• Lattice (order)

References

• Abramovich, Yuri A.; Aliprantis, C. D. (2002). An Invitation to Operator Theory. Graduate Studies in Mathematics. 50. American Mathematical Society. ISBN 0-8218-2146-6.