I have been reading a textbook on wavelets. I faced the following statement,
Let us relax the condition that $A_s$ and $W_s$ be orthogonal to each other and assume that the wavelet $\psi_{k,s}\in W_s$ has a dual, $\tilde{\psi}_{k,s}\in \tilde{W}_s$. Duality implies that the biorthogonality condition is satisfied; namely
$$ \langle\psi_{k,j}, \tilde{\psi}_{m,l}\rangle=\delta_{k,m} \delta_{j,l} \hspace{3mm} j,k,l,m \in \mathbb{Z} $$
If $F$ denotes field we know that an inner product space is a vector space $V$ over the field $F$ together with an inner product, that is a map $\langle \cdot ,\cdot \rangle :V\times V\to F$.
As you can see in the definition of the inner product both functions are from the same space $V$ but in the inner product above $\psi_{k,j}$ is from $W_s$ and $\tilde{\psi}_{m,l}$ is from $\tilde{W}_s$ which is dual space of $W_s$. How is this possible?(It is not compatible with the definition of inner product). Since, $\tilde{\psi}_{m,l}$ is from the dual space it must be a linear functional which means $\tilde{\psi}_{m,l}:W_s \rightarrow \mathbb{R}$.
I cannot properly justify what's happening above with Riesz Representation Theorem.