(Solved): Problem 4. Consider the vector space \( \mathbb{C}^{n} \) consisting of all \( n \)-tuples of compl ...

Problem 4. Consider the vector space \( \mathbb{C}^{n} \) consisting of all \( n \)-tuples of complex numbers. (a) Prove that \[ \left\langle\left(a_{1}, a_{2}, \ldots, a_{n}\right),\left(b_{1}, b_{2}, \ldots, b_{n}\right)\right\rangle=\sum_{k=1}^{n} a_{k} \overline{b_{k}} \] is an inner product on \( \mathbb{C}^{n} \). (b) Prove that \[ \sum_{k=1}^{n}\left|a_{k}\right|\left|b_{k}\right| \leq \sqrt{\sum_{k=1}^{n}\left|a_{k}\right|^{2}} \sqrt{\sum_{k=1}^{n}\left|b_{k}\right|^{2}} \]