Therese Belivet—if that's even her real name—is our nineteen-year-old protagonist. And yes, Therese Belivet is her real name, but that's about all we know about her, because that's about all she knows about herself.
At the beginning of the novel, Therese is young, she hates her job, and she wants more out of life, though she doesn't know what. Basically, she's everyone between the ages of eighteen and forty-nine. Therese fumbles around trying to find herself, and she thinks about the past a little bit, like a nun who once told her, "You must learn to trust people, Therese. Remember that" (1.10).
So whom does she trust? Hot people, that's who. How do we know? Well, Therese attempts a friendship with Mrs. Robichek, but it doesn't work out. You might think this is because Mrs. Robichek is too lonely for Therese to tolerate, but we think it's that she's too ugly since Carol is lonely, too, but hot, and Therese tolerates her just fine.
To be fair to Mrs. Robichek, it could also be that Carol is a more dominant personality. Therese grew up getting life advice from a nun because she grew up in an orphanage, even though she wasn't an orphan. She was a half-orphan, if that's a thing, with a dead dad and mom she didn't like. So part of Therese is drawn to Carol is that she wants a strong female figure in her life. It's something she's never really had, and it isn't something Mrs. Robichek offers.
Therese sheds a bit of her skin to find her true personality beneath. She has an almost bohemian attitude toward money and personal possessions, even though she still waffles back forth, saying on one page that she doesn't have enough money to go on a trip, and on the next saying that she isn't worried about money at all. Ultimately, though, she gravitates toward treating money and things like they're of no real value.
Money and objects are symbolic for her. She breaks a statue during a fight with Richard—a metaphoric shattering of their connection—and she sells a necklace he gave her to symbolize the end of their relationship. Getting rid of these objects makes the end of their relationship permanent for her, like girls in movies who cut their hair after traumatic events. Oh yeah, Therese gets a new haircut at the end, too.
Therese also pays the bill at restaurants when Carol isn't around, because Carol always insists on paying. It's Therese's small way of rebelling against Carol's controlling nature and a way to equalize the two women. Therese often has to think of money, and Carol never does. Therese wants money to seem like no object to her, either.
At times, Carol can be condescending to Therese. Their age difference is both the primary reason for the condescension, and the reason Therese puts up with it. Therese is young and she's submissive, so when Carol says things like, "You're much too young to know your own mind. Or what you're talking about" (13.75), or "I mean, you're so young, Therese. You'll change. You'll forget" (22.100), or she pets her neck as she might pet "a dog" (13.111), Therese never argues or pushes back. Plus, Carol is right: Therese is young, and she doesn't know her own mind. In many ways, hers is a coming of age story.
Carol says twice that Therese is like a person "flung out of space" (4.55 and 15.198), but she never explains what she means by this. Perhaps she means that Therese is spacey. Therese changes her mind more than Hamlet. Her boyfriend, Richard, even buys her two pastries because she is notorious for being unable to choose.
The reason Therese can't make decisions is that she doesn't know what she likes. That's why she's with Richard—she's doing what society expects her to do, not what she wants to do. And, to put it bluntly, what she wants to do is Carol.
Therese starts the process of finding herself by imitating Carol—she often "tr[ies] to sound as detached as Carol" (6.39), and she works through her feelings in letters to Carol that she doesn't send. By the end of the book, Therese becomes a little more like Carol. Importantly, though, Carol also becomes a little more like Therese, losing her domination over the dominant role in their relationship. By the end, Therese is the more dominant one, the one making decisions for the two women. When Therese returns Carol's car, it's Therese in the driver's seat—literally and metaphorically.