Does it make any difference when finding the adjoint operator, from which spaces they act. This example:

Question

I have this example: $$A(x_1,x_2,..x_n,...)=(\lambda_1 x_1,\lambda_2 x_2,...,\lambda_n x_n)$$ For: $ \ \\ a.) \ \ A:c_0 \to c_0 \\ b.) \ \ A: l_2 \to l_2$

I think this makes no difference because using the equality: $$<Ax,y^{*}>=<x,A^{*}y^{*}>$$ I feel you can find the adjoint of this operator, completely independant of what the spaces from which the operator acts upon are. (in this case for example, and in many that I have come across). Thoughts? How do the spaces play a role? There respective norms?