In $C[a,b]$, define the product
$\langle f,g\rangle= \int f(t)g(t) \, dt $.
Show that this product satisfies the property, $\langle f,f\rangle$ is greater than zero for all non zero $f$ using a non graphical proof and no δs to explain your answer.
Although this question was claimed to be a duplicate of another question, I would like an explanation without δs.
In the context of Linear Algebra, it's obvious that $\int_a^b f^2(x)dx$ is a positive quantity for a continuous nonzero $f(x)$. So as a Linear Algebra question, it's a bit off-topic.
For an analysis person, it's a little different. A "proof" would go like this. If $f(x)$ is continuous, so is $f^2(x)$. If $f^2(x)$ is nonzero, then there is a point where it's positive. It is known for continuous functions that if it's positive at a point, then it's positive over some interval that contains that point - there's a common $\epsilon\text{-}\delta$ argument that proves that. Over that interval, the integral is positive and it is therefore positive over $\left[a,b\right]$.