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Definition D6037
Euclidean real component sum operation

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 $$\mathbb{R}^{N \times 1} \to \mathbb{R}, \quad x \mapsto \boldsymbol{1}^T x$$

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 $$\mathbb{R}^N \to \mathbb{R}, \quad x \mapsto \sum_{n = 1}^N x_n$$