= Solution
Use the convention that the <inner product> is linear in its first argument. By the defining identity for the <adjoint operator>,
$$
\begin{aligned}
x\in(\operatorname{Im}T^*)^\perp
&\Longleftrightarrow\langle x,T^*y\rangle=0\quad\text{for every }y\in X\\
&\Longleftrightarrow\langle Tx,y\rangle=0\quad\text{for every }y\in X\\
&\Longleftrightarrow Tx=0.
\end{aligned}
$$
For the last implication choose $y=Tx$. This proves the first <image-kernel orthogonality for an adjoint> identity. Taking <orthogonal complements> and using the <double orthogonal complement> theorem for the <vector subspace> $\operatorname{Im}T^*$ gives
$$
\boxed{\ker T=(\operatorname{Im}T^*)^\perp,\qquad(\ker T)^\perp=\overline{\operatorname{Im}T^*}.}
$$
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