Question about the proof that the orthogonal complement of the kernel is the image of the adjoint

Question

Let $V$ be an inner product space, and let $T:V \rightarrow V$ be a linear operator such that the adjoint $T^*:V \rightarrow V$ exists. Prove that $\operatorname{im} T^*=(\ker T)^\perp$.

If I know the fact that $(\operatorname{im}T^*)^\perp=\ker T$, then from $(W^\perp)^\perp=W$, we can automatically lead to the result. Is that right?

Answer 1

Yes, this is essentially right. Note that you also need that $\ker T$ is a closed subspace since $(W^\perp)^\perp = W$ is only true if the subspace $W$ is closed.