Short answer: There is no solid evidence to back the claims behind Prevagen. Like most supplements, there is no need to provide FDA-approved studies in order to make general claims.
Longer answer: The active ingredient in Prevagen is apoaequorin, a calcium binding protein found in jellyfish. Quincy Bioscience, the manufacturer of Prevagen, claims that “The unique ingredient in Prevagen, apoaequorin...has been clinically shown to be safe and improve memory and support brain function.” They cite a self-sponsored study as the basis for their claims, but it is not peer-reviewed. Many other scientists view the study as biased and flawed. The Federal Trade Commission took action against Quincy Bioscience, stating that “Defendants have committed acts and practices that constitute repeated and persistent fraudulent and illegal conduct and false advertising in violation of NY GBL §§ 349 and 350, including misrepresenting, directly or indirectly, expressly or by implication, that Prevagen improves memory; Prevagen is clinically shown to improve memory...” There are serious questions about how trustworthy Quincy Bioscience is, and given the lack of empirical proof, it is unlikely that it significantly improves brain function.