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One of the answers in this link gives an existence proof for the adjoint operator

Existence is straightforward to establish if you have an orthonormal basis, say $e_k$:

Then, with $v= \sum_k v_k e_k$, we have $\langle f(v), w \rangle = \sum_k \overline{v_k} \langle f(e_k), w \rangle = \langle \sum_k v_k e_k, \sum_k \langle f(e_k), w \rangle e_k \rangle$, that is, $\langle f(v), w \rangle = \langle v, w' \rangle $, where $w'=f^*(w) = \sum_k \langle f(e_k), w \rangle e_k $.

I don't understand how this works if the vector spaces are different, because you could otherwise not equate the right sides of the inner product, $f^*(w) = \sum_k \langle f(e_k), w \rangle e_k$.

Is there something I'm missing that lets you do this. How would you prove this if the inner products were different for the vector spaces?

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  • $\begingroup$ It's not clear what problem you think there is here (there isn't actually any problem). Can you say more precisely what step you think is wrong and why? $\endgroup$ – Eric Wofsey Mar 26 '18 at 5:13

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