Let $\mathbb{R}^{N \times 1}$ be a D5656: Set of real column matrices.
The component sum operation on $\mathbb{R}^{N \times 1}$ is the D4364: Real function \begin{equation} \mathbb{R}^{N \times 1} \to \mathbb{R}, \quad x \mapsto \boldsymbol{1}^T x \end{equation}

Let $\mathbb{R}^N$ be a D5630: Set of euclidean real numbers.
The component sum operation on $\mathbb{R}^N$ is the D4364: Real function \begin{equation} \mathbb{R}^N \to \mathbb{R}, \quad x \mapsto \sum_{n = 1}^N x_n \end{equation}