I'm trying to understand a proof of the descent lemma, which says that if f is a continuously differentiable function over $\mathbb{R}^n$ with L-Lipschitz continuous gradient. Then, for any $x, y \in \mathbb{R}^n$:
$f(y) \leq f(x) + \nabla f(x)^T(y-x)+ \frac{L}{2}||x-y||^2$
By using the Fundamental Theorem of Calculus:
$\int_a^b f(x) dx = F(b) - F(a) \qquad F' = f$
And by defining:
$ g(t) = f(x + t(y-x)) $
Which gives:
$g(0) = f(x) \qquad g(1)=f(y)$
The proof starts by using the Fundamental Theorem of Calculus to write:
$f(y) - f(x) = \int_0^1 \langle \nabla f(x+t(y-x)), y-x\rangle dt $
However I don't get why there's a inner product in the integration, to my understanding shouldn't it be:
$f(y) - f(x) = \int_0^1 \nabla f(x+t(y-x)) (y-x) dt $
Your notation is unclear. Anyway, by the chain rule, $$ \frac{d}{dt} f(x+t(y-x)) = \nabla f(x+t(y-x)) \cdot (y-x), $$ where the dot denotes the inner product.