Definitions of Reflexivity

Question

I have met reflexive with two definitions:

1. If $A$ is a normed space, then $A$ is reflexive whenever $A=A^{**}$.

2. An operator algebra $A$ is reflexive if the algebra can be recovered from its invariant subspace lattice $L = Lat(A)$ as the set $Alg(L)$ of all operators leaving each subspace invariant.

I'm confused. Is there any relation between two definitions?