''Proof" is proof that you can drain most of the juice out of a play and still have an enjoyable night at the movies. The film comes with all sorts of Good Housekeeping seals of approval: David Auburn's stage drama won a Pulitzer and a Tony in 2001, Sir Anthony Hopkins is on board as the ghost of the heroine's father, and John Madden is directing Gwyneth Paltrow for the first time since he helped her win an Oscar for ''Shakespeare in Love." You don't need to wear evening dress to see the movie, but it couldn't hurt.
On stage, Auburn's play was a one-set, two-act marvel concerning Catherine (here played by Paltrow), the morose, bedraggled daughter of a recently deceased college mathematics professor named Robert (Hopkins). Daddy was a genius in youth and a paranoid schizophrenic for most of his adult life, and Catherine is worried she may be next. She has spent so long caring for her father in their rattletrap off-campus house, and so little time caring about her social life or physical appearance, that she may as well be a genius herself.
And so it turns out to be. Maybe. One of her father's students, a handsome, self-confessed math geek named Hal (Jake Gyllenhaal), comes across a handwritten proof among the professor's effects that may rock the academic world. We're never told exactly what it is because we're, you know, dumb, but there's every possibility that it may be the work of the daughter rather than the father. Can the insistence of an unstable young woman be taken as its own proof? Or do matters of faith and trust enter into the picture?
''Proof," in other words, is a glib and well-written inquiry into Big Things, and on that level it's good entertainment. The whole genius-equals-madness calculus is older than old hat, and so is the dad-was-crazy-and-I-may-be-too corollary -- Katharine Hepburn was treading these creaky boards in 1932's ''A Bill of Divorcement." But even with that Pulitzer, the play was never really about the ideas. It's about characters and acting, and Paltrow honors that.
Her Catherine is more strident, more melodramatically messed up than was Mary Louise Parker's nearly adorable basket case on Broadway -- Paltrow's is a movie performance, and she gives every tremor of anger and doubt its own close-up. The rest of the cast stands back and lets her swing the cat: Gyllenhaal's Hal is conciliatory and concerned, like an old-fashioned suitor with a high GPA, and Hope Davis isn't able to make Catherine's humorless sister Claire more than the dramatic convention she is. (Someone, please, break this actress out of the ghetto of purse-lipped supporting shrews.) Paltrow's the whole show, her eyes puffing with resentful tears and her nasal, patrician voice fraying at the edges. To her credit, she doesn't glamorize Catherine. To be blunt, she doesn't have to -- she's Gwyneth Paltrow.
Madden opens the play up as best he can -- he gets the cast off that front porch and into cars, living rooms, clothing boutiques -- but it's still a yakfest; not that there's anything wrong with that. Filmmaker Rebecca Miller (Arthur's daughter) has also been drafted to help Auburn with the script, to lesser effect: A line like ''I feel like I could crack open, like an egg, or one of those really smelly French cheeses that ooze when you cut them" has Miller's gummy fingerprints all over it.
In the end, ''Proof" isn't about proof or trust or mathematics or family so much as it's about one deeply alienated woman trying to decide whether she wants back into life. That's the drama of it, and Paltrow supplies her own proof.
Ty Burr can be reached at firstname.lastname@example.org.