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Let $f$ and $g$ be square-integrable, $\mathbb{R}^n$-valued functions, i.e., $$ \| f \|_2^2 = \int \|f(t)\|^2 dt < \infty $$ where $\|\cdot\|$ is the Euclidean norm on $\mathbb{R}^n$. I am looking for a nice proof of the vector-valued version of Cauchy-Schwarz, $$ \langle f,g \rangle_2^2 = \left( \int f^{T}(t) g(t) dt \right)^2 \leq \langle f,f \rangle_2 \langle g,g \rangle_2 = \int \|f(t)\|^2 dt \int \|g(t)\|^2 dt $$ using the CS inequality for scalar-valued functions on L2 and the CS inequality for vectors on $\mathbb{R}^n$.

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You can use the 'scalar' version of the Cauchy-Schwarz inequality more or less directly:

You have $f^T g \le \|f\| \|g\|$ for any $f,g \in \mathbb{R}^n$.

Hence $|\int f^T(t) g(t) dt |\le \int \|f(t)\| \|g(t)\| dt \le \sqrt{\int \|f(t)\|^2dt} \sqrt{\int \|g(t)\|^2dt}$

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