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Let $A\in B(\mathbb{C}^n) \cong \mathbb{M}_n(\mathbb{C} )$

Prove: The matix of $A^*$ (Banach space adjoint of $A$) with respect to the dual basis coincides with $A^t$ (the transpose of $A$).

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Let $(e_1^*,\ldots,e_n^*)$ the dual basis of $\Bbb C^n$ corresponding to the canonical basis $(e_1,\ldots,e_n)$. We denote $a_{lr}$ the entries of $A$. It's enough to compute $A^*$ such that $$\langle A^*e_j^*,e_k\rangle=\langle e_j^*,Ae_k\rangle.$$

Note that the result is $A^t$, and we don't take the conjugate.

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