# Gradient of an Inner Product in a more general Vector Space

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.

• I do not see what is more general in your question compared with the linked one. – daw Oct 12 '14 at 16:08
• taking the gradient and taking the derivative wrt to one variable are different things ... Right? – Charlie Parker Oct 12 '14 at 17:03
• w might be in a vector space of functions, is what I believe makes my question more general. Intuitively, it has more dimension I guess. – Charlie Parker Oct 12 '14 at 17:46