The phrase "a weak unit and" is grammatically correct and can be used in written English.
It can be used when describing a unit that lacks strength or effectiveness, often in a comparative or evaluative context.
Example: "In the analysis, we found that a weak unit and its performance metrics were significantly below the expected standards."
Alternatives: "a feeble unit and" or "an ineffective unit and".
On the other hand, if E is a p-convex Banach lattice with the σ-Fatou property and has a weak unit belonging to E a, then E is order isomorphic to a space L w p with ν on a σ-algebra [29].
If the existence of a weak unit is not assumed, the result remains true but for ν defined on a δ-ring.
This result generalizes [[29], Theorem 4] where E has a weak unit in E a. Remark that our proof differs from the one given in [[29], Theorem 4].
Depending on the fact that either the lattice contains a weak unit or not, either vector measures on σ-algebras or on δ-rings must be used.
Curbera proved in [[26], Theorem 8] that an order continuous Banach lattice F with a weak unit is always order isometric to a space L 1 , where ν is defined on a σ-algebra.
Remark that ℓ q cannot be written as an L p for a vector measure β defined on a σ-algebra since the space has not a weak unit.
Again, the corresponding result in the case when F has not a weak unit can be established by using a vector measure defined on a δ-ring as Delgado and Juan proved in [[27], Theorem 10].
When there exists also a weak unit in F, the Banach lattice can be represented with an L p -space but with ν defined on a σ-algebra (see [[20], Proposition 2.4]).
Thinking now about the space L w 1 , Curbera and Ricker showed in [[28], Theorem 2.5] that every Banach lattice F with the σ-Fatou property, having a weak unit which belongs to the σ-order continuous part F a of F, is order isometric to a space L w 1 with ν defined on a σ-algebra.
However, the team's new front-office executives have taken a new approach in rebuilding what was a weak unit.
