In The Taming of the Shrew, is the woman Katherina being tamed?
To understand whether or not Kathrine is "tamed", it is necessary to understand her and her world.
Kathrine lives in a male dominated society where a female was told who to marry by her father. As demonstrated from the suitors we see, it is not difficult to understand why she behaves as she does. Better to be considered a shrew than marry Gremio. Bianca is willing to be controlled by her father and then by her husband (chosen by her father) and Kathrine is not.
Petruchio reconizes the uniqueness of Kathrine when they meet and verbally spar. She is unlike any woman he has ever met and worth the challenge. Kathrine has played her role for a long time and does not easily yield but when they are on the way to her sister's weding and meet old Vencentio, she finally understands her husband's game and playfully joins him. By the time they get to the wedding, they are a team.
One could say that she is tamed when she comes on Petruchio's demand in Act V but it could be that the two of them are playing a game, as it were, with the others. How serious is her speech? I would contend that she really means what she says. In a sense she is saying her wedding vows. Earilier they were forced from her in the church but she really means what she is saying since she and Petruchio have found a level ground of love and understanding. Theirs will be a wonderful marrige full of verbal battles and fun. One can't say the same for the other marriages in the play. Both new brides have already shown their true colors.
Tamed? I think not. Found her match? Yes.
I took a slightly different reading of your question and laughed, yet again, at the possibility that it is Petruchio that has been tamed and not Katherina at all.
I agree that Katherina absolutely has altered her behavior and is no longer irate and ill-tempered all of the time. Perhaps that is being tamed. Oh, but I always enjoy going so much further than that by saying that it is Petruchio who has truly learned to love another and, through this intelligence, has been able to temper Katherina to the fullest. Even eNotes summary describes Petruchio (at the early part of the play) as "slovenly and vulgar." An avid description, to be sure. However, by the end of the play (and even amid tests of her compliance) Katherina is invited by Petruchio to share in his jokes, such as "mistaking" Vincentio as a young woman. Further, I submit that some of his tests are actually moments when he engages in word play with Katherina (such as the latter sun/moon incident) so as to make her an absolute equal. So, if some of the greatest joys of women of the time are to earn praise, continue in high society, receive love and bounteous affection (which I would beg to differ a bit, but I also don't live during Shakespeare's time), then it is Petruchio who gives in, and not Katherina.
Then again, perhaps I am far too influenced by the old 1980's Moonlighting version of the play which, by the way, if you haven't seen, you must. It is truly a classic.
My first answer would be "Yes, but…" By that I mean, yes, Katherina is "tamed" to a degree. She modifies her behavior. Simply put, she isn't as mean or loud, and she doesn't fight everything and everyone at the end of the play. However, if you mean is she domesticated, or is her spirit broken, I don't think so. I think Petruchio's done a tiny bit of taming, but for the most part, he's shown how far he's willing to go to have her as his wife, and this shows that she's valued. He also shows he's her match, rather than being just another person he can dominate.