On the difference between Malliavin derivative and Gross-Sobolev derivative

Let $W=C_0([0,1],mathbb R^d)$ be the classical Wiener space equipped with $mu$ the Wiener measure.
If $F:Wtomathbb R$ is a cylindrical function of the form

begin{align*}
F(w)=f(W_{t_1}(w),cdots,W_{t_n}(w)), finmathcal S(mathbb R^n)
end{align*}
where $W_t$ is the $t$-th component projection, we define for $hin H$ (the Cameron-Martin space),

begin{align*}
nabla_h F(w)=frac{d}{depsilon}F(w+epsilon h)rvert_{epsilon=0}
end{align*}

Notice that $W_t(w+h)=W_t(w)+h(t)$ and then

begin{align}
nabla_h F(w)&=frac{d}{depsilon}f(W_{t_1}(w+epsilon h),cdots,W_{t_n}(w+epsilon h))rvert_{epsilon=0}\
&=sum_{i=1}^n partial_i f(W_{t_1}(w+epsilon h),cdots,W_{t_n}(w+epsilon h))) h(t_i) rvert_{epsilon=0}\
&=sum_{i=1}^n partial_i f(W_{t_1}(w),cdots,W_{t_n}(w)) h(t_i).
end{align}

At this point we can see that the map $hmapsto nabla_h F(w)$ defines a random variable taking values in the dual space $H^*$, hence if we identify $H$ with its dual we have that $nabla F$ is an $H$-valued random variable.
If we extend this operator to all Wiener functionals (using the density of the cylindrical ones) we obtain the so called "Gross-Sobolev" derivative.

This definition looks pretty similar to the definition of the Malliavin derivative, but I don’t believe that the Mallivin derivative is a random variable taking values in the Cameron Martin space.

My idea on how things go (I am not completely sure and that’s why I am asking you this) is the following:

The last expression above can be written as

begin{align}
&=sum_{i=1}^n partial_i f(W_{t_1}(w),cdots,W_{t_n}(w)) int_0^1 1_{[0,t_i]}(s)dot{h}(s)ds\
&=sum_{i=1}^n partial_i f(W_{t_1}(w),cdots,W_{t_n}(w)) langle 1_{[0,t_i]},dot{h}rangle_{L^2(mu)}\
&=langle DF,dot{h}rangle_{L^2(mu)}\
&=bigglangle int_0^{cdot}D_rF dr,hbiggrangle_{H}=langle nabla F,hrangle_H.
end{align}

Where $DF$ is the classical Malliavin derivative of $F$. Then if what I’ve done above is correct the Gross-Sobolev derivative equals the time integral of the Malliavin derivative.
Can we thing of them as being the same since the space $L^2(mu)$ is isometric and isomorphic to $H$?

Is this correct, or is it just a coincident?

Thanks in advance for any feedback.