In the filed of interpolation spaces, given a compatible couple $(X_0,X_1)$ of Banach spaces i.e. $X_0$ and $X_1$ are embedded into a topological vector space, Peetre's $K$ and $J$ functionals are defined by for $x \in X_0 + X_1$, $$K(t,x) := \inf (\|x_0\|_{X_0} + t \|x_1\|_{X_1})$$ where infimum is taken all the decomposition $x = x_0 + x_1$ with $x_0 \in X_0$ and $x_1 \in X_1$, and for $x \in X_0 \cap X_1$, $$J(t,x) := \max (\|x\|_{X_0}, t \|x\|_{X_1}).$$
What is the motivation or intuitive meaning for the $K$ and $J$ functionals? What was the intention behind these functionals? In most books, these functionals are defined without explanation.