Let $(x_n)$, $(y_n) \in \mathbb{R}^\infty$ be sequences.
Is it possible to define an inner product $\langle \cdot , \cdot \rangle$ whereby $\langle (x_n) , (y_n) \rangle = c, c \in \mathbb{R}$?
I am asking this question because while this operation is natural for vectors of arbitrarily finite dimensions, I have never seen it being done for sequences.
Can we possibly take as definition:
$\langle (x_n) , (y_n) \rangle = x_1y_1 + x_2y_2 + \ldots + x_ny_n + \ldots$?
On
Well, a reason why you have never seen it for sequences is that, extending the existing formula for the regular inner product, we would have the following: $$\langle(x_n),(y_n)\rangle=\sum_{k=1}^\infty x_ky_k$$ which is an infinite sum, so there is mouch doubt about its convergence - we do not want any vector of our space $v$ with $\langle v,v\rangle=\pm\infty$ or, even worse, $\langle v,u\rangle$ not defined for some $u,v$.
To avoid such problems, we introduce the following spaces: $$l^p:=\left\{x\in\mathbb{R}^\infty\left|\sum_{k=1}^\infty|x_k|^p<+\infty\right.\right\}$$ One can show that $l^p$ is a normed space for every $p\geq1$ (and, actually, for $p=+\infty$), but that only for $p=2$ there is an inner product that generates the norm of that space. So, in the case of $l^2$, the inner product of two sequences $x,y\in l^2$ is: $$\langle x,y\rangle=\sum_{k=1}^\infty|x_ky_k|$$ from which we can take also the formula of the norm: $$\lVert x\rVert^2=\langle x,x\rangle=\sum_{k=1}^\infty|x_k|^2$$ Note that this infinite sum is convergent, since $x\in l^2$. For further reading you can read also this link.
If you are only interested in some inner product and don't care what properties are satisfied by the norm/metric/topology then the question involves only the cardinality. Take a vector space isomorphism from your space onto $l^2$ for example and define the inner product by taking the inner product of the images. Of course, this doesn't give a formula for the inner product.