Partial Derivatives along an Orthonormal Basis of a Function Differentiable on a Hilbert Space

In the setting of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation, with the orthonormal basis $(e_{k})_{k\in\mathbb{N}}$ of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §space. Differentiability on $X$ and the gradient $D\varphi(x)$ are those of that definition, applied with $E=U=X$, which is an open subset of itself by Real Hilbert Spaces: Standing Notation and Background §topology.

(Partial derivatives along the basis)¶ Let $\varphi:X\to\mathbb{R}$ be differentiable on $X$ and $k\in\mathbb{N}$. The $k$-th partial derivative of $\varphi$ is the function $\partial_{k}\varphi:X\to\mathbb{R}$, $\partial_{k}\varphi(x)=\langle D\varphi(x),e_{k}\rangle$.