I was looking at the following question:
Differentiating an Inner Product
that was talking about the derivative of an inner product to be:
$$ \frac{d}{dt} \langle f, g \rangle = \langle f(t), g^{\prime}(t) \rangle + \langle f^{\prime}(t), g(t) \rangle $$
However, I was looking for a more general answer when $f$ and $g$ belong to a more general Vector Space. In that case does the equivalent conjecture hold?
$$ \bigtriangledown_w \langle w, x \rangle = \langle w, \bigtriangledown_wx \rangle + \langle \bigtriangledown_ww, x \rangle $$
I believe the above does not hold and that it needs some slight modification to make it work. Does someone know what this modification might be? What is the gradient of a more general inner product.
