Inner Products, Norms & Angles

Let $V$ be a real vector space. An inner product (or scalar product) is a map $\langle\cdot,\cdot\rangle:V\times V\to\mathbb{R}$ that assigns a real number to a pair of vectors and satisfies four axioms for all $u,v,w\in V$ and all $\lambda\in\mathbb{R}$ (Definition 8.1): (1) additivity in the first slot, $\langle u+v,w\rangle=\langle u,w\rangle+\langle v,w\rangle$; (2) homogeneity in the first slot, $\langle\lambda u,v\rangle=\lambda\langle u,v\rangle$; (3) symmetry, $\langle u,v\rangle=\langle v,u\rangle$; and (4) positive definiteness, $\langle u,u\rangle\ge 0$ with $\langle u,u\rangle=0\iff u=0$. A space carrying such a map is an inner product space. Because of symmetry, linearity in the first slot automatically yields linearity in the second ($\langle u,v+w\rangle=\langle u,v\rangle+\langle u,w\rangle$ and $\langle u,\lambda v\rangle=\lambda\langle u,v\rangle$), so an inner product is bilinear. Inner products need not be the dot product: on $\mathbb{R}^2$ the weighted formula $\langle v,w\rangle=2v_1w_1+5v_2w_2$ is a genuine inner product (Example 8.1), since it is bilinear, symmetric, and $\langle v,v\rangle=2v_1^2+5v_2^2>0$ whenever $v\neq 0$.

A norm $\|\cdot\|:V\to\mathbb{R}$ measures length. By Definition 8.2 it satisfies $\|\lambda v\|=|\lambda|\,\|v\|$ (scaling), the triangle inequality $\|u+v\|\le\|u\|+\|v\|$, and $\|v\|=0\iff v=0$. Theorem 8.1 says every inner product induces a norm by $\|v\|=\sqrt{\langle v,v\rangle},$ which is well defined because positive definiteness makes $\langle v,v\rangle\ge 0$. The only delicate axiom is the triangle inequality, and it follows from the Cauchy–Schwarz inequality (Lemma 8.1), which in norm notation reads $|\langle u,v\rangle|\le\|u\|\,\|v\|$. The central example is the dot product on $\mathbb{R}^n$, $u\cdot v=u^Tv=u_1v_1+\cdots+u_nv_n$ (Definition 8.3), which is an inner product (Theorem 8.2); its induced norm is the Euclidean norm $\|v\|=\sqrt{v\cdot v}=\sqrt{v_1^2+\cdots+v_n^2}$ (Definition 8.4), sometimes written $\|\cdot\|_2$. Not every norm comes from an inner product, however: the taxicab norm $\|v\|_1=|v_1|+\cdots+|v_n|$ and the maximum norm $\|v\|_\infty=\max_i|v_i|$ are perfectly good norms with no underlying inner product (Remark 8.1).

With the dot product in hand, Definition 8.5 imports the familiar Euclidean geometry into $\mathbb{R}^n$. Two vectors are orthogonal (perpendicular), written $u\perp v$, exactly when their dot product vanishes: $u\perp v\iff u\cdot v=0.$ The length (or magnitude) of $v$ is $\|v\|=\sqrt{v\cdot v}$, and $v$ is a unit vector when its length is $1$, equivalently $v\cdot v=1$. The distance between two vectors is the length of their difference, $d(u,v)=\|u-v\|$. Two facts are worth memorising (Remark 8.2): the zero vector is orthogonal to every vector, since $0\cdot v=0$; and any nonzero $v$ can be normalised into a unit vector pointing the same way, $\hat u=\frac{1}{\|v\|}\,v.$ For instance $v=(3,4)^T$ has $\|v\|=\sqrt{9+16}=5$, so its unit vector is $\left(\tfrac35,\tfrac45\right)^T$.

Expanding a squared norm with bilinearity gives $\|x+y\|^2=(x+y)\cdot(x+y)=\|x\|^2+2(x\cdot y)+\|y\|^2$. The cross term $2(x\cdot y)$ disappears precisely when $x\perp y$, which is the Pythagorean theorem (Theorem 8.3): $\|x+y\|^2=\|x\|^2+\|y\|^2\iff x\perp y.$ Cauchy–Schwarz also lets us define angles. For nonzero $x,y$ it guarantees $-1\le\dfrac{x\cdot y}{\|x\|\,\|y\|}\le 1$, so there is a unique angle $0\le\theta\le\pi$ with $\cos\theta=\frac{x\cdot y}{\|x\|\,\|y\|}$ (Definition 8.6); orthogonal vectors meet at $\theta=90^\circ$, where the cosine is $0$. For example $x=(1,0,1)^T$ and $y=(0,1,1)^T$ give $x\cdot y=1$ and $\|x\|=\|y\|=\sqrt2$, so $\cos\theta=\tfrac12$ and $\theta=60^\circ$. Remark 8.3 cautions that both length and angle depend on the chosen inner product — switch to a weighted inner product and the same two vectors generally subtend a different angle.