To the naked eye, one could assume that Undine Spragg changed when she turned from a midwestern rich girl into a New York City socialite with links to the most prestigious families.
In a closer look we realize that Undine Spragg's character is actually flat and does not change at all throughout the story. For a character to be round and dynamic it must change its behavior, or its expectations, as a result of the changes it experiences.
However, Undine's character does not experience any change as a consequence of all her bad choices. Her only goal remained throughout the story: To become a New York socialite. When she switched her interests from having a prestigious name to having money, she did it with the same intention of remaining in New York's upper stratum.
After she failed to persuade Peter Van Degen to marry her she did not change into a more humble and less ambitious woman. She may have slowed down, but her wish was stronger than ever. In the end we see how, even with all the riches that her new husband had, she cringed at the thought of someone other than her husband working for the foreign office. Once again Undine's mind went to her "what if" mode and, once again, she showed a deep dissatisfaction with her situation.
This shows that Undine would always be a desperate, ambitious, and dissatisfied woman. She began and ended the story as one. And, from what we can infer, she will never change at all.