Fix constants $ w_1,...,w_n > 0 $, and for $ x,y \in \mathbb{R}^n $, define: $$ \big < x,y \big >_w = \sum_{k=1}^n w_ix_iy_i \text{.} $$ Verify that this yields an inner product on $ \mathbb{R}^n $. How would we need to modify this definition for it to yield an inner product on $ \mathbb{C}^n $? What about $ l^2(\mathbb{N}) $?
I only have a question about the inner product on $ l^2 $. Would I need to add conditions like $ w_n $ converges as well as $ x_n = y_n $ and $ x_n $ converges in $ l^2$ to make it work?
For $\mathbb{C}^n$:
$$\big < x,y \big >_w = \sum_{i=1}^n w_ix_i\overline{y}_i,$$
where $\overline{y_i}$ denotes the complex conjugate of $y_i$.
For $\mathcal{l}^2(\mathbb{N})$:
$$\big < x,y \big >_w = \sum_{i=1}^{+\infty} w_ix_iy_i.$$
To ensure convergence, a necessary condition is that $w_i \to 0$ when $i \to +\infty$.