2. For \( F=\mathbf{R} \) or \( \mathbf{C} \), let \( V \subset F^{\infty} \) be the set of sequences \( \left(a_{n}\right)_{n=1}^{\infty} \) such that \( \sum_{n=1}^{\infty}\left|a_{n}\right|^{2} \) converges (i.e., such that the sequence \( \left\{\sum_{n=1}^{N}\left|a_{n}\right|^{2}\right\}_{N=1}^{\infty} \) converges in \( \left.\mathbf{R}\right) \). i) Prove that \( V \) is a vector subspace of \( F^{\infty} \). ii) Prove that \[ \langle a, b\rangle:=\sum_{n=1}^{\infty} a_{n} \overline{b_{n}}\left[:=\lim _{N \rightarrow \infty} \sum_{n=1}^{N} a_{n} \overline{b_{n}}\right] \]

To show that \( V \) is a vector subspace of \( F^{\infty} \), we need to verify the following properties: 1. The zero sequence is in \( V \): The zero sequence \( (0, 0, 0, \ldots) \) has the property that \( \sum_{n=1}^{\infty} |0|^2 = 0 \), which converges. Hence, the zero sequence is in \( V \). 2. Closure under addition: Let \( (a_n) \in V \) and \( (b_n) \in V \). This means \( \sum_{n=1}^{\infty} |a_n|^2 < \infty \) and \( \sum_{n=1}^{\infty} |b_n|^2 < \infty \). By the Cauchy-Schwarz inequality, we can conclude that \( \sum_{n=1}^{\infty} |a_n + b_n|^2 \leq \sum_{n=1}^{\infty} |a_n|^2 + \sum_{n=1}^{\infty} |b_n|^2 < \infty \). Thus, \( (a_n + b_n) \in V \). 3. Closure under scalar multiplication: For any \( (a_n) \in V \) and \( c \in F \), we need to show \( (c a_n) \in V \). We have \( \sum_{n=1}^{\infty} |c a_n|^2 = |c|^2 \sum_{n=1}^{\infty} |a_n|^2 < \infty \). Therefore, \( (c a_n) \in V \). Thus, \( V \) is indeed a vector subspace of \( F^{\infty} \). Now for the inner product definition: We define the inner product \( \langle a, b \rangle := \sum_{n=1}^{\infty} a_n \overline{b_n} \). This sum converges under the condition that both \( (a_n) \) and \( (b_n) \) belong to \( V \). By properties of series and limits, we can interchange the summation and the limit when established convergence is guaranteed, leading to \( \langle a, b \rangle = \lim_{N \rightarrow \infty} \sum_{n=1}^N a_n \overline{b_n} \) which confirms that the inner product is well-defined with respect to this structure. This inner product satisfies linearity in the first argument, conjugate symmetry, and positivity, making it a suitable inner product for this space.