Question: Let \({{z}_{1}},...,{{z}_{n}}\) and \({{w}_{1}},...,{{w}_{n}}\) be complex numbers

(a) Show that

\[\left| \sum\limits_{k=1}^{n}{{{z}_{k}}{{{\bar{w}}}_{k}}} \right|\le \sum\limits_{k=1}^{n}{|{{z}_{k}}||{{w}_{k}}|}\]

(b) If \(\sum\limits_{k=1}^{n}{|{{z}_{k}}{{|}^{2}}=1}\) and \(\sum\limits_{k=1}^{n}{|{{w}_{k}}{{|}^{2}}=1}\), then show that

\[\sum\limits_{k=1}^{n}{|{{z}_{k}}{{{\bar{w}}}_{k}}|\,\le \sqrt{\sum\limits_{k=1}^{n}{|{{z}_{k}}{{|}^{2}}}\sum\limits_{k=1}^{n}{|{{w}_{k}}{{|}^{2}}}}}\]

(c) Let \(Z=\left( {{z}_{1}},...,{{z}_{n}} \right)\), and \(W=\left( {{w}_{1}},...,{{w}_{n}} \right)\) be two vectors in \({{\mathbb{C}}^{n}}\). Show that

\[\,\left| \sum\limits_{k=1}^{n}{{{z}_{k}}{{{\bar{w}}}_{k}}} \right|\overset{{}}{\mathop{\le }}\,\sqrt{\sum\limits_{k=1}^{n}{|{{z}_{k}}{{|}^{2}}}\sum\limits_{k=1}^{n}{|{{w}_{k}}{{|}^{2}}}}\]