Proof of ℝⁿ Is a Real Hilbert Space

Vector space. Let $x,y,z\in\mathbb{R}^{n}$ and $s,t\in\mathbb{R}$. The sum is associative and commutative, the origin satisfies $x+0=x$, and $x+(-x)=0$ (Additive Laws for Vectors in $\mathbb{R}^q$ and for Real $p\times q$ Matrices: Group Laws and Rules for Differences (i)). The operations being the pointwise ones on $\mathbb{R}^{[n]}$ (Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (iv)), the scalar laws

$$s(tx)=(st)x,\qquad 1x=x,\qquad t(x+y)=tx+ty,\qquad (s+t)x=sx+tx$$

hold by Rules for Pointwise Operations on Real-Valued Maps: a Commutative Ring, Scalar Multiples and Evaluation (iii). These are the axioms of Vector Space over a Field, so $\mathbb{R}^{n}$ is a real vector space (Vector Space over a Field (x)). Since the zero vector and negatives are unique (Uniqueness of the Zero Vector and of Negatives (i), Uniqueness of the Zero Vector and of Negatives (ii)), the zero vector is the origin $0$ and the negative of $x$ is the componentwise $-x$ (The Zero Vector, Negatives and Differences in a Vector Space (i), The Zero Vector, Negatives and Differences in a Vector Space (ii)); the difference of the vector space is then $x+(-y)$, which is the componentwise $x-y$ (The Zero Vector, Negatives and Differences in a Vector Space (iii), Additive Laws for Vectors in $\mathbb{R}^q$ and for Real $p\times q$ Matrices: Group Laws and Rules for Differences (ii)).

Inner product. The dot product is symmetric, additive and homogeneous in the first argument, and satisfies $x\cdot x\ge0$ with $x\cdot x=0$ only if $x=0$ (The Dot Product on $\mathbb{R}^n$ Is Symmetric, Bilinear and Positive Definite (i), The Dot Product on $\mathbb{R}^n$ Is Symmetric, Bilinear and Positive Definite (ii), The Dot Product on $\mathbb{R}^n$ Is Symmetric, Bilinear and Positive Definite (iv), The Dot Product on $\mathbb{R}^n$ Is Symmetric, Bilinear and Positive Definite (vi)). As $0$ is the zero vector, these are the conditions Real Inner Product Spaces (i), Real Inner Product Spaces (ii), Real Inner Product Spaces (iii) and Real Inner Product Spaces (iv). So the dot product is an inner product on $\mathbb{R}^{n}$ (Real Inner Product Spaces (v)), and $\mathbb{R}^{n}$ with it is a real inner product space (Real Inner Product Spaces (vi)). This proves (i).

Norm and distance. The induced norm of $x$ is $\sqrt{x\cdot x}$, which is the Euclidean norm $\lVert x\rVert$ (Real Inner Product Spaces (vii), The Euclidean Norm on $\mathbb{R}^n$ (i)). Since the difference of the vector space is the componentwise one, the norm metric is

$$d_V(x,y)=\lVert x-y\rVert=d_{E}(x,y)$$

(The Norm Metric of a Real Normed Space (i), The Euclidean Distance on $\mathbb{R}^n$ (i)). This proves (ii).

Completeness. $(\mathbb{R}^{n},d_{E})$ is complete ($\mathbb{R}^n$ with the Euclidean Distance Is a Complete Metric Space with Coordinatewise Convergence (iv)). By (ii) the norm metric is $d_{E}$, so the real normed space $\mathbb{R}^{n}$ is a real Banach space (Real Banach Space (i)), and $\mathbb{R}^{n}$ with the dot product is a real Hilbert space (Real Hilbert Spaces (i)). This proves (iii).