Can a continuous linear form have a norm of infinity?

Question

We know that a linear form $A \in V^{*}$ is continuous iff $$ \exists C: C\in R, ||Av|| \leq c ||v|| \forall v \in V $$ but we know too that $$ ||A|| = min\{c>0:||Av|| \leq C||v|| \forall v\in V\}$$ Does that mean that the norm of any continuous linear form is bounded since the set above must contain at least a real number? Sorry if the question seems trivial I'm just new to functional analysis

Answer 1

Yes, it is exactly like that. For linear operators between normed spaces, continuity and boundedness are equivalent.