Let $X$ be a Banach space and let $T:X \rightarrow X$ be a bounded linear map. Show that: If $T$ is surrjective then its transpose $T':X' \rightarrow X'$ is bounded below.
My try: We know that $R^\perp_M = N_{M'}$ and since X is surrjective $R_M = X$ hence $R_M^\perp = N_{M'} = 0$ so $M'$ is invertible and bounded below. Am I missing some details? Is invertible and bounded below true?