Inner Product Space

Author

John Robin Inston

Published

September 6, 2026

1 Inner Product

Given linear vector space \(\mathcal{V}\) an inner product \(\left< \cdot, \cdot \right>:\mathcal{V}\times\mathcal{V} \to \mathbb{R}\) is a mapping satisfying the following properties for all \(v_{1}, v_{2}, v_{3} \in \mathcal{V}\):

  1. Symmetry: \(\left< v_{1}, v_{2} \right> = \left< v_{2}, v_{1} \right>\);
  2. Non-negativity: \(\left< v_{1}, v_{1} \right> \geq 0\);
  3. Non-Degeneracy: \(\left< v_{1}, v_{1} \right> =0 \iff v_{1}=0\); and
  4. Linearity: \(\left< \alpha v_{1}, v_{2} + v_{3} \right> = \alpha \left< v_{1}, v_{2} \right> + \alpha \left< v_{1}, v_{3} \right> = \alpha \left< v_{1}, v_{2} + v_{3} \right>.\)

Note: Every inner product creates a norm satisfying: \(\lVert v \rVert = \sqrt{ \left< v, v \right> }.\)

2 Inner Product Space

A vector space with a corresponding inner product is known as an Inner Product Space.

An inner product space is said to be complete if every Cauchy Sequence \(\{v_{n}\}\) in the inner product space, there exists a limit \(v \in \mathcal{V}\) such that \(\lim_{n \to \infty} v_{n} = v\).

A complete inner product space is known as a Hilbert space.

2.1 Inner Product Space Results

Some important inner-product space results include:

  1. The Cauchy-Schwartz Inequality: \(\lvert \left< v_{1}, v_{2} \right> \rvert\leq \lVert v_{1} \rVert \cdot \lVert v_{2} \rVert\).
  2. The Triangle Inequality: \(\lVert v_{1}+v_{2} \rVert \leq \lVert v_{1} \rVert + \lVert v_{2} \rVert\).

3 Bilinear Forms

A function \(\langle\cdot,\cdot\rangle:\mathcal{V}\times\mathcal{V}\to F\) on a vector space \((\mathcal{V},F)\) is a bilinear form if it is linear in each argument: \(\langle\alpha\vec{x}+\vec{y},\vec{z}\rangle=\alpha\langle\vec{x},\vec{z}\rangle+\langle\vec{y},\vec{z}\rangle\) and \(\langle\vec{x},\alpha\vec{y}+\vec{z}\rangle=\alpha\langle\vec{x},\vec{y}\rangle+\langle\vec{x},\vec{z}\rangle\). It is symmetric if \(\langle\vec{x},\vec{y}\rangle=\langle\vec{y},\vec{x}\rangle\) and positive definite if \(\langle\vec{x},\vec{x}\rangle>0\) for all \(\vec{x}\neq\vec{0}\).

Every bilinear form on \(\mathbb{R}^n\) has the form \(\langle\vec{x},\vec{y}\rangle_A=\vec{x}^\top A\vec{y}\) for a unique \(A\in M_n(\mathbb{R})\), and conversely every such \(A\) defines one. The form is symmetric if and only if \(A\) is symmetric, and positive definite if and only if \(A\) is positive definite. Thus an inner product on a real vector space is exactly a symmetric positive definite bilinear form.

4 Euclidean Space

On \(\mathbb{R}^n\) the dot product \(\langle\vec{x},\vec{y}\rangle=\sum_{i=1}^nx_iy_i\) is an inner product (the matrix \(A=I_n\)). It induces the Euclidean norm \(\lVert\vec{x}\rVert=\sqrt{\langle\vec{x},\vec{x}\rangle}=\left(\sum_ix_i^2\right)^{1/2}\), the metric \(d(\vec{x},\vec{y})=\lVert\vec{x}-\vec{y}\rVert\), and the angle \(\theta\) between nonzero vectors via \(\cos\theta=\frac{\langle\vec{x},\vec{y}\rangle}{\lVert\vec{x}\rVert\lVert\vec{y}\rVert}\). The angle is well defined because of Cauchy-Schwarz below.

5 Cauchy-Schwarz and the Induced Norm

For any \(\vec{x},\vec{y}\) in an inner product space, \[ \lvert\langle\vec{x},\vec{y}\rangle\rvert\leq\lVert\vec{x}\rVert\,\lVert\vec{y}\rVert, \] with equality if and only if \(\vec{x},\vec{y}\) are linearly dependent.

Proof. If \(\vec{y}=\vec{0}\) both sides are \(0\). Otherwise let \(\vec{z}=\vec{x}-\frac{\langle\vec{x},\vec{y}\rangle}{\langle\vec{y},\vec{y}\rangle}\vec{y}\). Then \[ 0\leq\langle\vec{z},\vec{z}\rangle=\langle\vec{x},\vec{x}\rangle-\frac{\langle\vec{x},\vec{y}\rangle^2}{\langle\vec{y},\vec{y}\rangle}, \] which rearranges to the inequality. Equality holds iff \(\vec{z}=\vec{0}\), that is, \(\vec{x}\) is a multiple of \(\vec{y}\).

\(\lVert\vec{v}\rVert=\sqrt{\langle\vec{v},\vec{v}\rangle}\) is a norm, and it satisfies the parallelogram law \[ \lVert\vec{x}+\vec{y}\rVert^2+\lVert\vec{x}-\vec{y}\rVert^2=2\lVert\vec{x}\rVert^2+2\lVert\vec{y}\rVert^2. \]

Proof. Positivity and homogeneity are immediate. For the triangle inequality, by Cauchy-Schwarz \[ \lVert\vec{x}+\vec{y}\rVert^2=\lVert\vec{x}\rVert^2+2\langle\vec{x},\vec{y}\rangle+\lVert\vec{y}\rVert^2\leq(\lVert\vec{x}\rVert+\lVert\vec{y}\rVert)^2. \] The parallelogram law follows by expanding both squares, as the cross terms \(\pm2\langle\vec{x},\vec{y}\rangle\) cancel.

The parallelogram law characterises the norms that come from an inner product. For orthogonality, Gram-Schmidt and projections see Orthogonality and Projections.

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