Let $\mathcal{S}(\mathbb{R}^3,\mathbb{R}^3)$ be the space of vector fields on $\mathbb{R}^3$ with Schwartz function components.
Next, let $\mathcal{S}_{div}(\mathbb{R}^3,\mathbb{R}^3)$ be the space of divergence-free vector fields on $\mathbb{R}^3$ with Schwartz function components.
Then, the well-known Helmholtz decomposition tells us that the mapping \begin{equation} \mathbb{P} : \mathcal{S}(\mathbb{R}^3,\mathbb{R}^3) \to \mathcal{S}_{div}(\mathbb{R}^3,\mathbb{R}^3) \end{equation} is a projection, which is in fact known as the Leray-Hopf projection.
Now, fix any $f \in \mathcal{S}(\mathbb{R}^3,\mathbb{R}^3)$ which has nonzero divergence, so that $f \neq \mathbb{P}f$ as functions.
However, with the notation \begin{equation} \langle f, g \rangle : =\int_{\mathbb{R}^3} f \cdot g \end{equation} it is clear that \begin{equation} \langle f, g \rangle = \langle \mathbb{P} f, g \rangle \end{equation} for all $g \in \mathcal{S}_{div}(\mathbb{R}^3,\mathbb{R}^3)$, so that \begin{equation} f = \mathbb{P} f \text{ as elements of the dual space of } \mathcal{S}_{div}(\mathbb{R}^3,\mathbb{R}^3) \end{equation}
Is my reasoning correct? So, $f$ and $\mathbb{P}f$ do not coincide functions BUT coincide as continuous linear functionals on the space of divergence-free vector fields?
This looks quite subtle and confusing.. Could anyone pleae clarfiy?
Let $V$ be an inner product space with a subspace $W$ and an orthogonal projection $\pi:V\to W.$
Then, for $\mathbf{v}\in V\setminus W$ one has $\pi\mathbf v\neq \mathbf v$ but still $\langle \pi \mathbf v, \mathbf u\rangle = \langle \mathbf v, \mathbf u\rangle$ for all $\mathbf u\in W$ since $\mathbf v - \pi\mathbf v$ is orthogonal to $W.$
This is exactly your situation with $V=\mathcal{S}(\mathbb{R}^3,\mathbb{R}^3),$ $W=\mathcal{S}_{div}(\mathbb{R}^3,\mathbb{R}^3)$ and $\pi=\mathbb P.$