Professor von X
Something to Prove
Like Judith Shakespeare (check out her "Character Analysis"), Professor von X isn't real. He begins life as a picture Mary Beton draws while thinking about what all of the angry men who write books about women must be like. We're just going to say it: Professor von X is not a good-looking guy. (Here's a thought: maybe he's so angry because he can't find a girlfriend.)
And we're talking rage-a-hol levels of anger: he's "engaged in writing his monumental work entitled The Mental, Moral, and Physical Inferiority of the Female Sex" (2.11). That is a serious axe to grind. (And we bet he is the expert at mansplaining.)
One of Us
Just like Judith Shakespeare (and Mary Beton), Professor von X could be anyone and anywhere. Woolf tells us that straight up: "Nobody in their senses could fail to detect the dominance of the professor. He was the power and the money and the influence. He was the proprietor of the paper and its editor and sub-editor. He was the Foreign Secretary and the Judge. He was the cricketer..." (2.12).
Wait... wasn't Professor von X just a character Mary Beton made up? How could he suddenly be not just the professor writing a book, but every man in England?
Woolf is using the professor to illustrate the pervasiveness of sexism in her culture. Professor von X isn't really a foreign secretary, a judge, a cricket player, the paper owner, etc. He represents all the angry men in positions of power who feel better about themselves by insisting that women are inferior.