There is no such thing as a "linear value" per se, since linearity is on the arrow. But speaking of linear values makes sense when there's an ambient monad, and

There is no such thing as a "linear value" per se, since linearity is on the arrow. But speaking of linear values makes sense when there's an ambient monad, and all of the restrictions you expect are enforced; see Section 2.7 (and the last paragraph of Section 6.1) of https://arxiv.org/pdf/1710.09756.pdf https://arxiv.org/pdf/1710.09756.pdf for details.