This is merely about notations. In Conway's Functional Analysis textbook, exercise I.2.4 on p.11 to be precise, the closed linear span of a set $A\subset H$, where $H$ is a hilbert space, is defined as the intersection of all closed subspaces containing set $A$, and is denoted as $\vee A$.
However, in the subject of von Neumann algebras, there is the topic of lattice of projections. There, for two orthogonal projections $p,q$, along with their ranges $Y,Z\subset H$, which are closed subspaces, the greatest lower bound of $p$ and $q$ is denoted as $p\wedge q$, which is defined to be the orthogonal projection onto the intersection $Y\cap Z$. Here it is also customary to write the intersection as $Y \wedge Z$. This is consistent with the general definition of greatest lower bound in order theory.
So my question is, why using different symbols, $\vee$ and $\wedge$, in these two situations, when they both mean intersections of some closed subspaces, respectively? Is it just Conway's weird notation, or is there actually something deeper about this??
Thanks in advance.
The intersection of all closed subspaces containing $A$ is equal to the smallest closed subspace that contains $A$, hence some kind of infimum, which justifies the notation $\vee A$.