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Let’s consider the vector space \(\mathcal{C}^([a,b],\mathbb R)\) of continuous real functions defined on a compact interval \([a,b]\). We can define an inner product on pairs of elements \(f,g\) of \(\mathcal{C}^([a,b],\mathbb R)\) by \[ \langle f,g \rangle = \int_a^b f(x) g(x) \ dx.\] It is known that \(f \in \mathcal{C}^([a,b],\mathbb R)\) is the always vanishing function if we have \(\langl...
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