I am trying to prove the convergence of the inner space in $\mathbb{C}^{\infty}$ Defined as $$ \langle a | b \rangle := \sum_{k=1}^{\infty}b_{k}^{*}a_{k} $$ Considering that every sequence satisfy $$||a||^{2}= \sum_{k=1}^{\infty} |a_k|^2 < \infty $$ Here is my attempt, By cauchy schwarz inequality, $$ \left| \sum_{k=1}^{\infty} b_{k}^{*} a_{k} \right|^{2} ≤ ( \sum_{k=1}^{\infty} \left| b_{k} ^{*} \right|^{2} ) ( \sum_{k=1}^{\infty} \left| a_{k} \right|^{2}) $$ From the condition of the norm, there is a $M \in \mathbb{R} $ such that, $$ \left| \sum_{k=1}^{\infty} b_{k}^{*} a_{k} \right| < M $$ It can be seen from the last argument that the series has a lower and upper bound. I get stuck there as i can not find a way to prove convergence from the previous idea.
I would appreciate if anyone can give me a hint.
Suppose $a_k$ and $b_k$ are positive, then by your analysis $\sum a_kb_k = \sum b_k^*a_k$ has an upper bound. The partial sums $\sum_{k=1}^n a_k b_k$ is also monotonous, so the whole series must converges.
Now get back to the general situation. For $x=(a_k)$ and $y=(b_k)$, $\sum b_k^*a_k$ is absolutely convergent. (Or, you can pass from the Cauchy criterion of series). So it is convergent.