how does gilman convey the belief prevalent in her time that women were emotionally unstable and prone to her ilness

According to the social norms of the time period, women in the 19th century were expected to fulfill their duties as wives and mothers and be content in their existence as nothing more. Men and women were divided between the public and private sphere, and women were doomed to spend their lives solely in the domestic sphere. Not coincidentally, women who dared to enter the masculine public realm were viewed as something akin to prostitutes, the lowest level of society. I think the idea that the narrator was somehow prone to emotional problems because she is a woman is meant to be ironic. The author wants readers to see through patriarchal ideas of "rest cure" and "hysteria" that were labels on women prevalent at the time.