I am totally stuck and have no idea whatsoever on how to prove the following inequality (by the way this is a problem from an undergraduate book in multivariable advanced calculus at Junior/Senior level ):
Let $g=\left ( g_{1},g_{2},...,g_{n} \right ): \left [ a,b \right ]\rightarrow \mathbb{R}^{n}$ is a continuous function, then we define: $\int_{a}^{b}g\left ( x \right )dx=\left \langle \int_{a}^{b}g_{1}\left ( x \right )dx,...,\int_{a}^{b}g_{n}\left ( x \right ) \right \rangle$
Prove that: $\left \| \int_{a}^{b}g\left ( x \right )dx \right \|\leq \int_{a}^{b}\left \| g\left ( x \right ) \right \|dx$
In the book, there is a hint saying that I should use the Cauchy Schwarz inequality, but I have no clue how to use it. The only I was able to prove is:
Left hand side= $\sqrt{\left (\int_{a}^{b}g_{1}\left ( x \right )dx \right )^{2}+...+\left ( \int_{a}^{b}g_{2}\left ( x \right )dx \right )^{2}}$
Right hand side is= $\int_{a}^{b}\sqrt{\left (g_{1}\left ( x \right ) \right )^{2}+...+\left ( g_{n}\left ( x \right ) \right )^{2}}dx$
I am looking forward for your suggestions and answers.