I have sometimes overheard people using the terms hard analysis and soft analysis.I am not a particularly well-read person in mathematics but I have wondered what that is all about.I hope there exists an explanation for someone with single-variable calculus background.
Roughly speaking, if you use "functional analysis" methods it is called soft, whereas if you use "estimates" it is called hard.
For example, Weierstrass constructed an example of a function $f \colon [0,1] \to \mathbb R$ that is continuous but nowhere differentiable. His proof involved computing inequalities to show his function was not differentiable. HARD
Nowadays a modern mathematician may consider the Banach space $C[0,1]$ and cite the Baire category theorem to show that there is a function $f \colon [0,1] \to \mathbb R$ which is continuous but nowhere differentiable. SOFT