The definition of coercivity and boundedness of a linear operator $L$ between two $B$ spaces looks similar: $\lVert Lx\lVert\geq M\_1\lVert x\rVert$ and $\lVert Lx\rVert\leq M\_2\lVert x\rVert$ for some constants $M\_1$ and $M\_2$. Thus in order to show the existence of a PDE $Lu=f$ one needs to show that it is coercive. However if my operator $L$ happen to be bounded and $M\_2 \leq M\_1$?

What is the intuition behind those two concepts because they are based on computation of the same quantities and comparing the two?