Isabel Archer is the protagonist in the novel and the "Lady" of the title. She is a young woman from Albany, New York, who travels to Europe with her Aunt, Mrs. Touchett. One of the central conflicts that is explored through the book is her desire for personal independence and to journey and explore and her commitment to social propriety. This is explored chiefly through her relationships with men. She refuses her persistent suitor Caspar Goodwood, then proceeds to reject an English peer, and finally falls prey to a villainous scheme to have her married to the questionable Gilbert Osmond.
You would do well to examine how Isabel Archer is manipulated and affected by the actions of others. Her cousin's decision to give her a large part of his inheritance seems to be very indulgent, especially as it made her the victim of Madame Merle and Osmond. Ralph Touchett's desire to give her the money so she can have her dreams realised indicates a voyeuristic nature, and we wonder whether he regrets his decision at the end of the novel. What Isabel is famous for though is her decision to stay in her unhappy marriage to Osmond and reject searching for a happier future.
Let's start with some surface-level details about Isabel Archer. When the reader is first introduced to Isabel, she is a young woman of 23. She's an American from Albany, New York. Her mother died when she was young. A consequence of that death is the fact that Isabel's father had to raise her, and he raised her to be an intelligent, well-read, and fiercely independent woman. If I had to pick one character trait of Isabel's that stands out more than any other character trait, it would be her independence. Isabel knows what she likes and doesn't like. She wants to do what she wants to do, and she doesn't want to be beholden to anything or anyone else. Isabel knows this about herself and admits it freely to other people.
"You meant she has taken me up. Yes; she likes to take people up. She has been very kind to me; but," she added with a certain visible eagerness of desire to be explicit, "I'm very fond of my liberty."
Isabel is so outspoken about her independence that other people easily notice it as one of her defining characteristics.
"Well, if you'll be very good, and do everything I tell you I'll take you there," Mrs. Touchett declared.
Our young woman's emotion deepened; she flushed a little and smiled at her aunt in silence. "Do everything you tell me? I don't think I can promise that."
"No, you don't look like a person of that sort. You're fond of your own way; but it's not for me to blame you."
Unfortunately, Isabel's independent spirit causes her to frequently push away legitimate suitors.
"Don’t think me unkind if I say it’s just that – being out of your sight – that I like. If you were in the same place I should feel you were watching me, and I don’t like that – I like my liberty too much. If there’s a thing in the world I’m fond of," she went on with a slight recurrence of grandeur, "it’s my personal independence."
The frequency by which Isabel seems to have gentlemen calling on her and asking for her hand in marriage tells me that Isabel is likely just as beautiful as she is intelligent and independent. Both Goodwood and Warburton propose to Isabel before she becomes wealthy, so the wealth she could bring into a marriage is not a concern at that point.
Perhaps because Isabel is young, talented, smart, attractive, and highly desired, she is also guilty of a bit of pride.
She had no talent for expression and too little of the consciousness of genius; she only had a general idea that people were right when they treated her as if she were rather superior. Whether or no she were superior, people were right in admiring her if they thought her so; for it seemed to her often that her mind moved more quickly than theirs, and this encouraged an impatience that might easily be confounded with superiority. It may be affirmed without delay that Isabel was probably very liable to the sin of self-esteem; she often surveyed with complacency the field of her own nature; she was in the habit of taking for granted, on scanty evidence, that she was right; she treated herself to occasions of homage.
What's interesting about Isabel's pride, though, is that she knows about it, yet she feels that it is entirely justified. That pride in herself is what makes the ending of the book so sad. I believe that Isabel's pride is also what makes her so committed to sticking with things. She knows, and readers know, that her marriage to Osmond is a sham. She knows that she is being used for her wealth, yet she made a vow and believes she needs to honor it by being the best wife that she can be.
"You won’t confess that you’ve made a mistake. You’re too proud."
"I don’t know whether I’m too proud. But I can’t publish my mistake. I don’t think that’s decent. I’d much rather die."
"You won’t think so always," said Henrietta.
"I don’t know what great unhappiness might bring me to; but it seems to me I shall always be ashamed. One must accept one’s deeds. I married him before all the world; I was perfectly free; it was impossible to do anything more deliberate."