ℝⁿ Is a Real Hilbert Space

In the setting of Euclidean Space and Real Matrices: Standing Notation, let $n\in\mathbb{N}$.

(i) Inner product space. $\mathbb{R}^{n}$, with its componentwise sum $x+y$ and scalar multiple $tx$, is a real vector space, whose zero vector, negatives and differences are the origin $0$, $-x$ and $x-y$ of $\mathbb{R}^{n}$. The dot product is an inner product on it, so $\mathbb{R}^{n}$ with the dot product is a real inner product space.

(ii) Norm and distance. The induced norm of this real inner product space is the Euclidean norm $\lVert\cdot\rVert$, and its norm metric, as a real normed space (The Norm of a Real Inner Product Space (iii)), is the Euclidean distance $d_{E}$.

(iii) Hilbert space. $\mathbb{R}^{n}$ with the dot product is a real Hilbert space.