Problem: Let $X$ be a Banach space and $H$ be a Hilbert space. Let $A$ be a linear continuous operator from $X$ to $H$. Here, we say that $A$ is coercive if there exists real number $c>0$ such that $$|| Ax||_H\geq c||x||_X, \hspace{0.5cm} \forall x\in X.$$ My question is under what condition, above coercivity holds?
Special case: Let us assume that $X=\mathbb R^n$ and $H=\mathbb R^m$ and $A$ is a matrix of size $m\times n$. If the symmetric matrix $A^TA$ is non-singular then we can choose $c=\lambda_1>0$ where $\lambda_1$ is the smallest eigenvalue of $A^TA$.