Problem 2
Verify that $$ \left(\sum_{k=1}^{n} a_{k} b_{k}\right)^{2}=\sum_{k=1}^{n} a_{k}^{2} \sum_{k=1}^{n} b_{k}^{2}-\frac{1}{2} \sum_{i=1}^{n} \sum_{j=1}^{n}\left(a_{i} b_{j}-b_{i} a_{j}\right)^{2} $$ Deduce the Cauchy-Schwarz inequality (3) from this identity.
Problem 3
Verify that $$ \left(\int_{a}^{b} x(t) y(t) d t\right)^{2}=\int_{a}^{b} x^{2}(t) d t \int_{a}^{b} y^{2}(t) \mathrm{dt}-\frac{1}{2} \int_{a}^{b} \int_{a}^{b}[x(s) y(t)-y(s) x(t)]^{2} d s \mathrm{dt} $$ Deduce Schwarz's inequality (11) from this identity.
