Coercivity vs boundedness of operator 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?
 A: With boundedness everything is clear, because it is well known that linear operator $L$ is continuous iff $L$ is bounded. With continuity of $L$ you can solve the equation with sequential approximations. Moreover, you can apply the whole theory developed for continuous functions and, in particular, for continuous linear operators. Since continuity is very natural condition when solving differential equations, we require $L$ to be bounded.
As for the coercivity, note that it, in particular, implies injectivity. Injectivity guarantees us uniqueness of the solution $u$ of the equation $Lu=f$. But when you are solving such an equation, it is desirable that solution depends on right hand side $f$ continuously. Well, this property depends on $L$, and it is sufficient to require coercivity of $L$. Speaking functional-analytically bounded coercive operator perform linear homeomorphism between domain and its range. Hence there is a "nice" correspondence between initial data $f$ and solution $u$.
A: The number $\inf_{x\neq 0}\frac{\lVert Lx\rVert}{\lVert L\rVert\cdot\lVert x\rVert}$ measure how injective $L$ is (when $L=0$ it's not well defined). 
Consider $R(x):=\lVert Lx\rVert$ for $x$ in the unit ball. Coercivity means that $R(x)\geq M_1>0$ for some constant $M_1$ and in particular $L$ is injective (but $L$ can be unbounded, for example if $B=\ell^2(\Bbb R)$ and $Le_n=ne_n$, $n\geq 1$).
A bounded operator doens't need to be coercive, for example $L\equiv 0$. 
To get an intuition, coercivity means that the vectors of the unit ball are map to a positive distance from $0$, and this distance is independent of the point. Boundedness measure how far form $0$ can be mapped the vectors of the unit ball. 
