Let $X$,$Y$ two finite-dimensional closed linear subspaces of $L^p := (L^p)^n$ (defined in a finite measure space). Define $$L^p_X = \{ f \in L^p : \int \langle f,x \rangle = 0 \quad \forall x \in X \},$$ where $\langle ,\rangle$ denotes the usual dot product of the $n$ components. I need to calculate the dimension of the kernel and the dimension of the cokernel (in terms of $dim(X)$ and $dim(Y)$) of the following applications:
$$ i: L^p_X / Y \to L^p_X$$ $$ pr: L^p_X \to L^p_X / Y$$ $$i': L^p_X \to L^p$$ $$pr': L^p \to L^p_X$$
where $i$, $i'$ are the natural inclusions, $pr$ is the natural projection and $$pr'(f)=f-\sum_i \left( \int \langle f, \varphi_i \rangle \right) \varphi_i$$
for $(\varphi_i)_i$ an orthonormal basis of $X$.
My attemp: It is pretty clear that $ker(i)$, $ker(i')$, $coker(pr)$ and $coker(pr')$ have dimension $0$. Also, it seems that $ker(pr)=Y$ and $ker(pr')=X$. For the cokernel of $i$ and $i'$, I need to "simplify" $L^p_X / (L^p_X / Y)$ and $(L^p/L^p_X)$. Are they isomorphic to $Y$ and $X$ respectively?
Another question is: Can I calculate the adjoints of the operators above? For example, is the natural projection $L^{p'}/X \to L^{p'}_Y/X$ the adjoint of $i$ above? What about the rest (especially $i'$ and $pr'$)?
I really want to understand how this stuff works. References are also welcomed. Thank you!