Inner Product Definition | Inner Product Spaces | Linear Algebra

1. Definition of Inner Product

An inner product equips a vector space with geometric structure—the ability to measure lengths and angles. We begin with the abstract definition, then explore concrete examples.

Definition 7.1: Inner Product (Real Vector Space)

Let $V$ be a vector space over $\mathbb{R}$ . An inner product on $V$ is a function $\langle \cdot, \cdot \rangle : V \times V \to \mathbb{R}$ satisfying:

Linearity in first argument: $\langle \alpha x + \beta y, z \rangle = \alpha\langle x, z \rangle + \beta\langle y, z \rangle$
Symmetry: $\langle x, y \rangle = \langle y, x \rangle$
Positive definiteness: $\langle x, x \rangle \geq 0$ , with equality iff $x = 0$

Remark 7.1: Bilinearity

For real inner products, symmetry plus linearity in the first argument implies linearity in the second argument:

\langle x, \alpha y + \beta z \rangle = \langle \alpha y + \beta z, x \rangle = \alpha\langle y, x \rangle + \beta\langle z, x \rangle = \alpha\langle x, y \rangle + \beta\langle x, z \rangle

Thus real inner products are bilinear (linear in both arguments).

Definition 7.2: Inner Product (Complex Vector Space)

Let $V$ be a vector space over $\mathbb{C}$ . An inner product on $V$ is a function $\langle \cdot, \cdot \rangle : V \times V \to \mathbb{C}$ satisfying:

Linearity in first argument: $\langle \alpha x + \beta y, z \rangle = \alpha\langle x, z \rangle + \beta\langle y, z \rangle$
Conjugate symmetry: $\langle x, y \rangle = \overline{\langle y, x \rangle}$
Positive definiteness: $\langle x, x \rangle \geq 0$ (real), with equality iff $x = 0$

Remark 7.2: Sesquilinearity

Complex inner products are sesquilinear (conjugate-linear in the second argument):

\langle x, \alpha y \rangle = \overline{\langle \alpha y, x \rangle} = \overline{\alpha}\overline{\langle y, x \rangle} = \bar{\alpha}\langle x, y \rangle

Note: Some texts use the opposite convention (linear in second argument). We follow the physicist's convention.

Remark 7.3: Why Conjugate Symmetry?

Conjugate symmetry ensures $\langle x, x \rangle$ is always real:

\langle x, x \rangle = \overline{\langle x, x \rangle} \implies \langle x, x \rangle \in \mathbb{R}

Without this, positive definiteness wouldn't make sense for complex spaces.

Definition 7.3: Inner Product Space

A vector space $V$ equipped with an inner product $\langle \cdot, \cdot \rangle$ is called an inner product space (or pre-Hilbert space).

Definition 7.3a: Hilbert Space

A Hilbert space is a complete inner product space—one where every Cauchy sequence converges. Finite-dimensional inner product spaces are automatically complete (and hence Hilbert spaces).

Remark 7.3a: Equivalent Formulations

The inner product axioms can be stated in several equivalent ways:

Alternative 1: Linearity in second argument, conjugate-linearity in first (mathematician's convention)
Alternative 2: Bilinear + symmetric + positive definite (real case only)
Alternative 3: Sesquilinear + Hermitian + positive definite

These are all equivalent up to which argument is linear vs. conjugate-linear.

Theorem 7.0: Inner Product from Positive Definite Matrix

For any positive definite Hermitian matrix $A$ , the function $\langle x, y \rangle_A = x^H A y$ defines an inner product on $\mathbb{C}^n$ .

Proof:

Linearity: $\langle \alpha x + \beta y, z \rangle_A = (\alpha x + \beta y)^H A z = \alpha x^H A z + \beta y^H A z$

Conjugate symmetry: $\langle y, x \rangle_A = y^H A x = (x^H A^H y)^* = (x^H A y)^* = \overline{\langle x, y \rangle_A}$

Positive definiteness: Since $A$ is positive definite, $x^H A x > 0$ for all $x \neq 0$ .

∎

Example 7.0a: Verifying Inner Product Axioms

Show that $\langle f, g \rangle = \int_0^1 f(x)g(x) \, dx$ is an inner product on $C[0,1]$ .

Linearity: $\langle \alpha f + \beta g, h \rangle = \int_0^1 (\alpha f + \beta g)h = \alpha \int_0^1 fh + \beta \int_0^1 gh$

Symmetry: $\langle g, f \rangle = \int_0^1 gf = \int_0^1 fg = \langle f, g \rangle$

Positive definiteness: $\langle f, f \rangle = \int_0^1 f^2 \geq 0$ , with equality iff $f = 0$ (for continuous $f$ ).

Remark 7.3b: Semi-Inner Products

If we relax positive definiteness to $\langle x, x \rangle \geq 0$ (allowing $\langle x, x \rangle = 0$ for $x \neq 0$ ), we get a semi-inner product or pseudo-inner product. This induces a seminorm rather than a norm.

Remark 7.3c: Historical Note

The concept of inner product was developed in the early 20th century. Key contributors include:

David Hilbert (1862-1943): Formalized infinite-dimensional spaces (Hilbert spaces)
John von Neumann (1903-1957): Axiomatized quantum mechanics using Hilbert spaces
Frigyes Riesz (1880-1956): Proved the Riesz representation theorem

Example 7.0b: Non-Example: Not an Inner Product

Consider $\langle x, y \rangle = x_1 y_1 - x_2 y_2$ on $\mathbb{R}^2$ .

For $x = (1, 1)^T$ : $\langle x, x \rangle = 1 - 1 = 0$ but $x \neq 0$ .

This violates positive definiteness, so it's NOT an inner product. (It's a symmetric bilinear form, but indefinite.)

Theorem 7.0a: Inner Product from Norm (Polarization)

If a norm $\|\cdot\|$ satisfies the parallelogram law, then it comes from an inner product defined by the polarization identity:

\langle x, y \rangle = \frac{1}{4}(\|x + y\|^2 - \|x - y\|^2)

for real spaces. For complex spaces, use the full polarization formula.

Remark 7.3d: Why Inner Products Matter

Inner products provide:

Geometry: Length, angle, orthogonality—making algebra geometric
Optimization: Projections give best approximations
Analysis: Completeness leads to Hilbert spaces and functional analysis
Physics: Quantum mechanics is built on complex Hilbert spaces
Applications: Signal processing, machine learning, statistics

2. Standard Examples

Example 7.1: Euclidean Inner Product on ℝⁿ

The standard (Euclidean, dot) inner product on $\mathbb{R}^n$ :

\langle x, y \rangle = \sum_{i=1}^{n} x_i y_i = x_1 y_1 + x_2 y_2 + \cdots + x_n y_n = x^T y

For $x = (1, 2, 3)^T$ and $y = (4, -1, 2)^T$ :

\langle x, y \rangle = 1(4) + 2(-1) + 3(2) = 4 - 2 + 6 = 8

Example 7.2: Standard Inner Product on ℂⁿ

The standard inner product on $\mathbb{C}^n$ :

\langle x, y \rangle = \sum_{i=1}^{n} x_i \overline{y_i} = x^* y = x^H y

where $x^H = \bar{x}^T$ is the conjugate transpose (Hermitian transpose).

For $x = (1+i, 2)^T$ and $y = (1, i)^T$ :

\langle x, y \rangle = (1+i)(1) + 2(-i) = 1 + i - 2i = 1 - i

Example 7.3: Weighted Inner Product

For positive weights $w_1, \ldots, w_n > 0$ , define on $\mathbb{R}^n$ :

\langle x, y \rangle_w = \sum_{i=1}^{n} w_i x_i y_i

This is useful when different coordinates have different importance or units.

Example 7.4: L² Inner Product on Function Spaces

On continuous functions $C[a,b]$ :

\langle f, g \rangle = \int_a^b f(t)g(t)\, dt

For $f(t) = t$ and $g(t) = t^2$ on $[0,1]$ :

\langle f, g \rangle = \int_0^1 t \cdot t^2\, dt = \int_0^1 t^3\, dt = \frac{1}{4}

Example 7.5: Inner Product on Polynomials

On $P_n(\mathbb{R})$ (polynomials of degree ≤ n):

\langle p, q \rangle = \int_{-1}^{1} p(x)q(x)\, dx

Legendre polynomials are orthogonal with respect to this inner product.

Example 7.6: Frobenius Inner Product on Matrices

On $M_{m \times n}(\mathbb{R})$ :

\langle A, B \rangle = \text{tr}(A^T B) = \sum_{i,j} a_{ij} b_{ij}

This treats a matrix as a vector of $mn$ entries.

Example 7.7: Matrix Inner Product (General)

For any positive definite matrix $A \in M_n(\mathbb{R})$ :

\langle x, y \rangle_A = x^T A y

This defines a valid inner product. When $A = I$ , we recover the standard inner product.

Example 7.7a: Sequence Space ℓ²

The space $\ell^2$ of square-summable sequences $(x_1, x_2, \ldots)$ :

\langle x, y \rangle = \sum_{n=1}^{\infty} x_n \overline{y_n}

This is an infinite-dimensional Hilbert space, fundamental in functional analysis.

Example 7.7b: Weighted L² Inner Product

For a positive weight function $w(t) > 0$ :

\langle f, g \rangle_w = \int_a^b f(t)g(t)w(t)\, dt

Different weights give different orthogonal polynomial families:

$w(t) = 1$ on $[-1,1]$ : Legendre polynomials
$w(t) = e^{-t}$ on $[0,\infty)$ : Laguerre polynomials
$w(t) = e^{-t^2}$ on $(-\infty, \infty)$ : Hermite polynomials
$w(t) = (1-t^2)^{-1/2}$ on $(-1,1)$ : Chebyshev polynomials

Example 7.7c: Inner Product on Complex Matrices

On $M_{m \times n}(\mathbb{C})$ , the Frobenius inner product:

\langle A, B \rangle = \text{tr}(A^H B) = \sum_{i,j} \overline{a_{ij}} b_{ij}

This induces the Frobenius norm: $\|A\|_F = \sqrt{\text{tr}(A^H A)} = \sqrt{\sum |a_{ij}|^2}$

Remark 7.4a: Classification of Inner Products

Inner products can be classified by their domain:

Finite-dimensional: $\mathbb{R}^n, \mathbb{C}^n$ , polynomial spaces $P_n$
Sequence spaces: $\ell^2, \ell^p$ (though $\ell^p$ for $p \neq 2$ is not an inner product space)
Function spaces: $L^2, C[a,b], H^1$ (Sobolev spaces)

3. Induced Norm

Definition 7.4: Induced Norm

Every inner product induces a norm (notion of length):

\|x\| = \sqrt{\langle x, x \rangle}

Theorem 7.1: Norm Properties

The induced norm satisfies:

Non-negativity: $\|x\| \geq 0$ , with $\|x\| = 0$ iff $x = 0$
Homogeneity: $\|\alpha x\| = |\alpha| \cdot \|x\|$
Triangle inequality: $\|x + y\| \leq \|x\| + \|y\|$

Proof:

(1) Follows directly from positive definiteness of inner product.

(2) $\|\alpha x\|^2 = \langle \alpha x, \alpha x \rangle = |\alpha|^2 \langle x, x \rangle = |\alpha|^2 \|x\|^2$

(3) Follows from Cauchy-Schwarz (proven next).

∎

Example 7.8: Euclidean Norm

On $\mathbb{R}^n$ with standard inner product:

\|x\| = \sqrt{x_1^2 + x_2^2 + \cdots + x_n^2}

For $x = (3, 4)^T$ : $\|x\| = \sqrt{9 + 16} = 5$

Example 7.9: L² Norm

On $C[a,b]$ :

\|f\|_2 = \sqrt{\int_a^b |f(t)|^2\, dt}

For $f(t) = \sin(t)$ on $[0, \pi]$ : $\|f\|_2 = \sqrt{\pi/2}$

Remark 7.4: Not All Norms Come from Inner Products

The 1-norm $\|x\|_1 = \sum |x_i|$ and ∞-norm $\|x\|_\infty = \max |x_i|$ do NOT come from any inner product. Only norms satisfying the parallelogram law (see below) are induced by inner products.

Theorem 7.1a: Distance Function

The induced norm defines a metric (distance function):

d(x, y) = \|x - y\| = \sqrt{\langle x - y, x - y \rangle}

This satisfies: (1) $d(x,y) \geq 0$ , (2) $d(x,y) = 0 \iff x = y$ , (3) $d(x,y) = d(y,x)$ , (4) $d(x,z) \leq d(x,y) + d(y,z)$ .

Example 7.9a: Computing Distances

In $\mathbb{R}^3$ with standard inner product, find the distance between $x = (1, 2, 3)^T$ and $y = (4, 0, 1)^T$ :

d(x,y) = \|x - y\| = \|(-3, 2, 2)\| = \sqrt{9 + 4 + 4} = \sqrt{17}

Example 7.9b: L² Distance Between Functions

The $L^2$ distance between $f(t) = t$ and $g(t) = t^2$ on $[0,1]$ :

d(f,g) = \sqrt{\int_0^1 (t - t^2)^2 dt} = \sqrt{\int_0^1 (t^2 - 2t^3 + t^4) dt}

= \sqrt{\frac{1}{3} - \frac{1}{2} + \frac{1}{5}} = \sqrt{\frac{10 - 15 + 6}{30}} = \sqrt{\frac{1}{30}}

Theorem 7.1b: Norm Squared Expansion

For any vectors $x, y$ :

\|x + y\|^2 = \|x\|^2 + 2\text{Re}\langle x, y \rangle + \|y\|^2

\|x - y\|^2 = \|x\|^2 - 2\text{Re}\langle x, y \rangle + \|y\|^2

Proof:

Expand using the definition of norm:

\|x + y\|^2 = \langle x + y, x + y \rangle = \langle x, x \rangle + \langle x, y \rangle + \langle y, x \rangle + \langle y, y \rangle

By conjugate symmetry: $\langle x, y \rangle + \langle y, x \rangle = \langle x, y \rangle + \overline{\langle x, y \rangle} = 2\text{Re}\langle x, y \rangle$

∎

Corollary 7.0a: Real Inner Products

For real inner product spaces:

\|x + y\|^2 = \|x\|^2 + 2\langle x, y \rangle + \|y\|^2

Remark 7.4b: Unit Vectors

A vector $x$ with $\|x\| = 1$ is called a unit vector. For any nonzero $x$ , the vector $\hat{x} = \frac{x}{\|x\|}$ is the normalization of $x$ (a unit vector in the same direction).

Example 7.9c: Normalizing a Vector

Normalize $x = (3, 4)^T$ :

\|x\| = \sqrt{9 + 16} = 5, \quad \hat{x} = \frac{1}{5}(3, 4)^T = (0.6, 0.8)^T

Verify: $\|\hat{x}\| = \sqrt{0.36 + 0.64} = 1$ ✓

4. Cauchy-Schwarz Inequality

Theorem 7.2: Cauchy-Schwarz Inequality

For any vectors $x, y$ in an inner product space:

|\langle x, y \rangle| \leq \|x\| \cdot \|y\|

Equality holds if and only if $x$ and $y$ are linearly dependent.

Proof:

If $y = 0$ , both sides are 0. Assume $y \neq 0$ .

For any scalar $t$ , positive definiteness gives:

0 \leq \|x - ty\|^2 = \langle x - ty, x - ty \rangle = \|x\|^2 - 2\text{Re}(t\langle x, y \rangle) + |t|^2\|y\|^2

Choose $t = \frac{\langle x, y \rangle}{\|y\|^2}$ (the optimal value):

0 \leq \|x\|^2 - \frac{|\langle x, y \rangle|^2}{\|y\|^2}

Rearranging: $|\langle x, y \rangle|^2 \leq \|x\|^2 \|y\|^2$

Taking square roots gives the result. Equality holds iff $x = ty$ for some scalar $t$ .

∎

Corollary 7.1: Triangle Inequality

\|x + y\| \leq \|x\| + \|y\|

Proof:

\|x + y\|^2 = \|x\|^2 + 2\text{Re}\langle x, y \rangle + \|y\|^2 \leq \|x\|^2 + 2|\langle x, y \rangle| + \|y\|^2

By Cauchy-Schwarz:

\leq \|x\|^2 + 2\|x\|\|y\| + \|y\|^2 = (\|x\| + \|y\|)^2

∎

Example 7.10: Cauchy-Schwarz in ℝⁿ

For $x = (1, 2, 3)^T$ and $y = (1, 1, 1)^T$ :

$\langle x, y \rangle = 1 + 2 + 3 = 6$
$\|x\| = \sqrt{14}$ , $\|y\| = \sqrt{3}$
$\|x\| \cdot \|y\| = \sqrt{42} \approx 6.48$

Indeed, $6 \leq 6.48$ ✓

Example 7.11: Cauchy-Schwarz for Functions

For $f, g \in L^2[a,b]$ :

\left| \int_a^b f(t)g(t)\, dt \right| \leq \sqrt{\int_a^b |f(t)|^2\, dt} \cdot \sqrt{\int_a^b |g(t)|^2\, dt}

Remark 7.5: Importance of Cauchy-Schwarz

The Cauchy-Schwarz inequality is one of the most important inequalities in mathematics:

Proves the triangle inequality for norms
Defines angles between vectors
Bounds correlations in probability
Proves Hölder's inequality (generalization)

Example 7.11a: Cauchy-Schwarz for Sums

For sequences $a_1, \ldots, a_n$ and $b_1, \ldots, b_n$ :

\left(\sum_{i=1}^n a_i b_i\right)^2 \leq \left(\sum_{i=1}^n a_i^2\right)\left(\sum_{i=1}^n b_i^2\right)

Example: $(1 \cdot 2 + 2 \cdot 1)^2 = 16 \leq (1 + 4)(4 + 1) = 25$ ✓

Theorem 7.2a: Alternative Proof via Discriminant

Consider the quadratic $q(t) = \|x + ty\|^2 = \|y\|^2 t^2 + 2\text{Re}\langle x,y\rangle t + \|x\|^2$ .

Since $q(t) \geq 0$ for all real $t$ , the discriminant must be non-positive:

4(\text{Re}\langle x,y\rangle)^2 - 4\|x\|^2\|y\|^2 \leq 0

This gives $|\text{Re}\langle x,y\rangle| \leq \|x\|\|y\|$ , and similarly for the imaginary part.

Corollary 7.1a: Equality in Cauchy-Schwarz

Equality $|\langle x, y \rangle| = \|x\| \cdot \|y\|$ holds if and only if:

$x = 0$ or $y = 0$ , or
$x = \lambda y$ for some scalar $\lambda$

In other words, equality holds iff $x$ and $y$ are linearly dependent.

Example 7.11b: When Equality Holds

For $x = (2, 4)^T$ and $y = (1, 2)^T = \frac{1}{2}x$ :

\langle x, y \rangle = 2 + 8 = 10, \quad \|x\| = \sqrt{20}, \quad \|y\| = \sqrt{5}

|\langle x, y \rangle| = 10 = \sqrt{20}\sqrt{5} = \|x\|\|y\|

Equality holds because $x = 2y$ (linearly dependent).

Theorem 7.2b: Reverse Cauchy-Schwarz

For nonzero vectors in a real inner product space:

\|x\| \cdot \|y\| \leq \|x + y\| \cdot \|x - y\| / |\cos\theta - \sin\theta|

when the denominator is nonzero, where $\theta$ is the angle between $x$ and $y$ .

Example 7.11c: Cauchy-Schwarz in Probability

For random variables $X, Y$ with finite second moments:

|E[XY]|^2 \leq E[X^2] \cdot E[Y^2]

This is Cauchy-Schwarz with $\langle X, Y \rangle = E[XY]$ , the $L^2$ inner product.

Equality holds iff $X$ and $Y$ are linearly related (one is a constant multiple of the other).

Corollary 7.1b: Correlation Bound

The correlation coefficient $\rho = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y}$ satisfies:

-1 \leq \rho \leq 1

This is a direct consequence of Cauchy-Schwarz applied to centered random variables.

Remark 7.5a: Generalizations

Cauchy-Schwarz generalizes to:

Hölder's inequality: $\|fg\|_1 \leq \|f\|_p \|g\|_q$ where $1/p + 1/q = 1$
Minkowski's inequality: Triangle inequality for $L^p$ norms
Bessel's inequality: Bounds on Fourier coefficients

Example 7.11d: Proving AM-GM via Cauchy-Schwarz

For positive $a_1, \ldots, a_n$ :

\frac{a_1 + \cdots + a_n}{n} \geq \sqrt[n]{a_1 \cdots a_n}

Apply Cauchy-Schwarz to $(\sqrt{a_1}, \ldots, \sqrt{a_n})$ and $(1/\sqrt{a_1}, \ldots, 1/\sqrt{a_n})$ .

5. Angles and Orthogonality

Definition 7.5: Angle Between Vectors

For nonzero vectors $x, y$ in a real inner product space, the angle $\theta$ between them is defined by:

\cos\theta = \frac{\langle x, y \rangle}{\|x\| \cdot \|y\|}

By Cauchy-Schwarz, $|\cos\theta| \leq 1$ , so $\theta \in [0, \pi]$ is well-defined.

Definition 7.6: Orthogonality

Vectors $x$ and $y$ are orthogonal (perpendicular), written $x \perp y$ , if:

\langle x, y \rangle = 0

This corresponds to $\theta = 90°$ (or $\pi/2$ radians).

Example 7.12: Orthogonal Vectors in ℝ³

Vectors $x = (1, 0, 1)^T$ and $y = (1, 0, -1)^T$ :

\langle x, y \rangle = 1(1) + 0(0) + 1(-1) = 0

So $x \perp y$ .

Example 7.13: Angle Calculation

For $x = (1, 1)^T$ and $y = (1, 0)^T$ :

\cos\theta = \frac{1 \cdot 1 + 1 \cdot 0}{\sqrt{2} \cdot 1} = \frac{1}{\sqrt{2}}

So $\theta = 45° = \pi/4$ .

Theorem 7.3: Pythagorean Theorem

If $x \perp y$ , then:

\|x + y\|^2 = \|x\|^2 + \|y\|^2

Proof:

\|x + y\|^2 = \langle x + y, x + y \rangle = \|x\|^2 + 2\langle x, y \rangle + \|y\|^2 = \|x\|^2 + \|y\|^2

since $\langle x, y \rangle = 0$ .

∎

Corollary 7.2: Generalized Pythagorean Theorem

If $x_1, \ldots, x_n$ are pairwise orthogonal:

\|x_1 + \cdots + x_n\|^2 = \|x_1\|^2 + \cdots + \|x_n\|^2

Definition 7.7: Orthogonal Complement

For a subset $S \subseteq V$ , the orthogonal complement is:

S^\perp = \{v \in V : \langle v, s \rangle = 0 \text{ for all } s \in S\}

Theorem 7.4: Orthogonal Complement is a Subspace

For any subset $S$ , $S^\perp$ is a subspace of $V$ .

Proof:

Zero vector: $\langle 0, s \rangle = 0$ for all $s$ , so $0 \in S^\perp$ .

Closure: If $x, y \in S^\perp$ and $\alpha, \beta$ are scalars:

\langle \alpha x + \beta y, s \rangle = \alpha\langle x, s \rangle + \beta\langle y, s \rangle = 0

∎

Example 7.13a: Orthogonal Vectors in Function Space

On $C[-\pi, \pi]$ with $\langle f, g \rangle = \int_{-\pi}^{\pi} f(x)g(x)\,dx$ :

The functions $\sin(nx)$ and $\cos(mx)$ are orthogonal for all integers $n, m$ :

\langle \sin(nx), \cos(mx) \rangle = \int_{-\pi}^{\pi} \sin(nx)\cos(mx)\,dx = 0

This orthogonality is the foundation of Fourier series.

Example 7.13b: Angle Between Functions

Find the angle between $f(x) = 1$ and $g(x) = x$ on $[-1, 1]$ :

\langle f, g \rangle = \int_{-1}^{1} 1 \cdot x\, dx = 0

Since $\langle f, g \rangle = 0$ , the angle is $\theta = 90°$ — they are orthogonal!

Theorem 7.4a: Orthogonality and Linear Independence

A set of nonzero pairwise orthogonal vectors is linearly independent.

Proof:

Suppose $\alpha_1 v_1 + \cdots + \alpha_n v_n = 0$ with $v_i \perp v_j$ for $i \neq j$ .

Take inner product with $v_k$ :

0 = \langle \sum \alpha_i v_i, v_k \rangle = \sum \alpha_i \langle v_i, v_k \rangle = \alpha_k \|v_k\|^2

Since $v_k \neq 0$ , we have $\|v_k\|^2 > 0$ , so $\alpha_k = 0$ .

∎

Corollary 7.2a: Maximum Orthogonal Set

In an $n$ -dimensional inner product space, any orthogonal set has at most $n$ nonzero vectors.

Definition 7.7a: Orthonormal Set

A set $\{e_1, \ldots, e_n\}$ is orthonormal if:

\langle e_i, e_j \rangle = \delta_{ij} = \begin{cases} 1 & i = j \\ 0 & i \neq j \end{cases}

That is, the vectors are pairwise orthogonal and each has unit length.

Example 7.13c: Standard Orthonormal Basis

The standard basis $\{e_1, \ldots, e_n\}$ of $\mathbb{R}^n$ is orthonormal:

e_1 = (1,0,\ldots,0)^T, \quad e_2 = (0,1,\ldots,0)^T, \quad \ldots

We have $\langle e_i, e_j \rangle = \delta_{ij}$ for the standard inner product.

Example 7.13d: Another Orthonormal Basis in ℝ²

The rotated basis $\{u_1, u_2\}$ where:

u_1 = \frac{1}{\sqrt{2}}(1, 1)^T, \quad u_2 = \frac{1}{\sqrt{2}}(1, -1)^T

Verify: $\|u_1\| = \|u_2\| = 1$ and $\langle u_1, u_2 \rangle = \frac{1}{2}(1 - 1) = 0$ .

Remark 7.6a: Coordinates in Orthonormal Basis

If $\{e_1, \ldots, e_n\}$ is orthonormal, coordinates are easy to find:

v = \sum_{i=1}^n \langle v, e_i \rangle e_i

The coefficient of $e_i$ is simply $\langle v, e_i \rangle$ —no system of equations needed!

Theorem 7.4b: Parseval's Identity

If $\{e_1, \ldots, e_n\}$ is an orthonormal basis and $v = \sum c_i e_i$ :

\|v\|^2 = \sum_{i=1}^n |c_i|^2 = \sum_{i=1}^n |\langle v, e_i \rangle|^2

The squared norm equals the sum of squared coefficients—a generalized Pythagorean theorem.

6. Parallelogram Law and Polarization

Theorem 7.5: Parallelogram Law

In any inner product space:

\|x + y\|^2 + \|x - y\|^2 = 2\|x\|^2 + 2\|y\|^2

Proof:

Expand both sides using the inner product:

\|x + y\|^2 = \|x\|^2 + 2\text{Re}\langle x, y \rangle + \|y\|^2

\|x - y\|^2 = \|x\|^2 - 2\text{Re}\langle x, y \rangle + \|y\|^2

Adding these gives the result.

∎

Remark 7.6: Geometric Interpretation

The parallelogram law states that the sum of the squares of the diagonals of a parallelogram equals the sum of the squares of all four sides. This is a fundamental property that characterizes inner product spaces.

Theorem 7.6: Characterization of Inner Product Norms

A normed vector space $(V, \|\cdot\|)$ is an inner product space (with the norm induced by that inner product) if and only if the norm satisfies the parallelogram law.

Example 7.14: 1-Norm Fails Parallelogram Law

On $\mathbb{R}^2$ with 1-norm, take $x = (1, 0), y = (0, 1)$ :

$\|x + y\|_1 = \|(1,1)\|_1 = 2$
$\|x - y\|_1 = \|(1,-1)\|_1 = 2$
LHS: $4 + 4 = 8$
RHS: $2(1) + 2(1) = 4$

Since $8 \neq 4$ , the 1-norm doesn't come from an inner product.

Theorem 7.7: Polarization Identity (Real Case)

In a real inner product space:

\langle x, y \rangle = \frac{1}{4}\left(\|x + y\|^2 - \|x - y\|^2\right)

Theorem 7.8: Polarization Identity (Complex Case)

In a complex inner product space:

\langle x, y \rangle = \frac{1}{4}\left(\|x + y\|^2 - \|x - y\|^2 + i\|x + iy\|^2 - i\|x - iy\|^2\right)

Remark 7.7: Significance of Polarization

The polarization identity shows that the inner product is completely determined by the norm. If you know all the lengths, you can compute all the inner products (and hence all the angles).

Example 7.14a: Using Polarization Identity

Given $\|x\| = 3, \|y\| = 4, \|x+y\| = 5$ in a real inner product space, find $\langle x, y \rangle$ :

\|x+y\|^2 = \|x\|^2 + 2\langle x,y\rangle + \|y\|^2

25 = 9 + 2\langle x,y\rangle + 16 \implies \langle x,y\rangle = 0

The vectors are orthogonal! (This is the 3-4-5 right triangle.)

Theorem 7.8a: Apollonius Identity

For any vectors $x, y$ and their midpoint $m = \frac{x+y}{2}$ :

\|x - z\|^2 + \|y - z\|^2 = 2\|m - z\|^2 + \frac{1}{2}\|x - y\|^2

This relates distances from a point $z$ to the endpoints and midpoint of a segment.

Proof:

Apply the parallelogram law to vectors $x - z$ and $y - z$ :

\|x-z\|^2 + \|y-z\|^2 = \frac{1}{2}\|(x-z)+(y-z)\|^2 + \frac{1}{2}\|(x-z)-(y-z)\|^2

= \frac{1}{2}\|x+y-2z\|^2 + \frac{1}{2}\|x-y\|^2 = 2\|m-z\|^2 + \frac{1}{2}\|x-y\|^2

∎

Remark 7.7a: Jordan-von Neumann Theorem

The parallelogram law completely characterizes inner product spaces among normed spaces. This deep result (Jordan-von Neumann, 1935) shows that the parallelogram law is the only additional axiom needed to get from a normed space to an inner product space.

Example 7.14b: Verifying the Parallelogram Law

In $\mathbb{R}^2$ with $x = (1, 2)^T, y = (3, 1)^T$ :

$\|x+y\|^2 = \|(4,3)\|^2 = 16 + 9 = 25$
$\|x-y\|^2 = \|(-2,1)\|^2 = 4 + 1 = 5$
LHS: $25 + 5 = 30$
$2\|x\|^2 + 2\|y\|^2 = 2(5) + 2(10) = 30$

Both sides equal 30 ✓

Theorem 7.8b: Polarization in Higher Dimensions

In a real inner product space, the polarization identity can be written as:

4\langle x, y \rangle = \|x+y\|^2 - \|x-y\|^2

In complex inner product spaces, we need all four terms:

4\langle x, y \rangle = \|x+y\|^2 - \|x-y\|^2 + i\|x+iy\|^2 - i\|x-iy\|^2

7. Important Properties

Theorem 7.9: Continuity of Inner Product

The inner product is continuous: if $x_n \to x$ and $y_n \to y$ (in norm), then:

\langle x_n, y_n \rangle \to \langle x, y \rangle

Proof:

|\langle x_n, y_n \rangle - \langle x, y \rangle| \leq |\langle x_n - x, y_n \rangle| + |\langle x, y_n - y \rangle|

By Cauchy-Schwarz:

\leq \|x_n - x\| \cdot \|y_n\| + \|x\| \cdot \|y_n - y\| \to 0

∎

Theorem 7.10: Reverse Triangle Inequality

\big| \|x\| - \|y\| \big| \leq \|x - y\|

Proof:

By triangle inequality: $\|x\| = \|(x-y) + y\| \leq \|x-y\| + \|y\|$

So $\|x\| - \|y\| \leq \|x-y\|$ . By symmetry, $\|y\| - \|x\| \leq \|x-y\|$ .

∎

Theorem 7.11: Properties of Orthogonal Complement

For subspaces $U, W$ of inner product space $V$ :

$U \subseteq W \implies W^\perp \subseteq U^\perp$
$U \cap U^\perp = \{0\}$
$U \subseteq (U^\perp)^\perp$
In finite dimensions: $V = U \oplus U^\perp$

Example 7.15: Orthogonal Complement in ℝ³

Let $U = \text{span}\{(1, 0, 0), (0, 1, 0)\}$ (xy-plane in $\mathbb{R}^3$ ).

Then $U^\perp = \text{span}\{(0, 0, 1)\}$ (z-axis).

And $\mathbb{R}^3 = U \oplus U^\perp$ .

Remark 7.8: Finite vs Infinite Dimensions

In finite-dimensional spaces, $U^{\perp\perp} = U$ always. In infinite dimensions, we only have $U \subseteq U^{\perp\perp}$ , with equality iff $U$ is closed.

Theorem 7.11a: Dimension Formula for Orthogonal Complement

For a subspace $U$ of finite-dimensional inner product space $V$ :

\dim(U) + \dim(U^\perp) = \dim(V)

Proof:

Since $V = U \oplus U^\perp$ (direct sum), dimensions add.

∎

Example 7.15a: Computing Orthogonal Complement

In $\mathbb{R}^3$ , let $U = \text{span}\{(1, 1, 0)^T\}$ . Find $U^\perp$ .

A vector $(x, y, z)^T \in U^\perp$ iff $\langle (x,y,z), (1,1,0) \rangle = x + y = 0$ .

So $U^\perp = \{(x, -x, z) : x, z \in \mathbb{R}\} = \text{span}\{(1,-1,0)^T, (0,0,1)^T\}$ .

Check: $\dim(U) + \dim(U^\perp) = 1 + 2 = 3$ ✓

Theorem 7.11b: Best Approximation Property

Let $U$ be a closed subspace of inner product space $V$ . For any $v \in V$ , there exists a unique $u \in U$ minimizing $\|v - u\|$ . This is characterized by:

v - u \in U^\perp

Remark 7.8a: Projection

The vector $u$ in the theorem is called the orthogonal projection of $v$ onto $U$ , written $P_U v$ or $\text{proj}_U v$ . It minimizes distance to $U$ .

Example 7.15b: Projection in ℝ³

Project $v = (1, 2, 3)^T$ onto the xy-plane $U = \{(x,y,0)\}$ :

P_U v = (1, 2, 0)^T

The residual $v - P_U v = (0, 0, 3)^T \in U^\perp$ (the z-axis).

Theorem 7.11c: Projection Formula

If $\{e_1, \ldots, e_k\}$ is an orthonormal basis for subspace $U$ :

P_U v = \sum_{i=1}^{k} \langle v, e_i \rangle e_i

Proof:

We need $v - P_U v \perp U$ . For any $e_j$ :

\langle v - \sum \langle v, e_i \rangle e_i, e_j \rangle = \langle v, e_j \rangle - \sum \langle v, e_i \rangle \langle e_i, e_j \rangle = \langle v, e_j \rangle - \langle v, e_j \rangle = 0

∎

8. Inner Products and Matrices

Theorem 7.12: Matrix Representation of Inner Products

Let $V$ be an $n$ -dimensional real inner product space with basis $\{e_1, \ldots, e_n\}$ . Define the Gram matrix:

G_{ij} = \langle e_i, e_j \rangle

Then for $x = \sum x_i e_i$ and $y = \sum y_j e_j$ :

\langle x, y \rangle = x^T G y

Definition 7.8: Positive Definite Matrix

A symmetric matrix $G \in M_n(\mathbb{R})$ is positive definite if:

x^T G x > 0 \quad \text{for all } x \neq 0

Equivalently, all eigenvalues of $G$ are positive.

Theorem 7.13: Characterization of Inner Product Matrices

$\langle x, y \rangle = x^T G y$ defines an inner product on $\mathbb{R}^n$ if and only if $G$ is symmetric and positive definite.

Example 7.16: Non-Standard Inner Product

Let $G = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$ . Check positive definiteness:

x^T G x = 2x_1^2 + 2x_1 x_2 + 2x_2^2 = x_1^2 + (x_1 + x_2)^2 + x_2^2 > 0

for $x \neq 0$ . So $\langle x, y \rangle = x^T G y$ is a valid inner product.

Remark 7.9: Standard Basis

For the standard inner product on $\mathbb{R}^n$ with standard basis, $G = I$ . A basis where $G = I$ is called an orthonormal basis.

Theorem 7.14: Hermitian Matrices for Complex Inner Products

On $\mathbb{C}^n$ , $\langle x, y \rangle = x^H G y$ defines an inner product iff $G$ is Hermitian ( $G = G^H$ ) and positive definite.

Example 7.17: Checking Hermitian Positive Definiteness

Is $G = \begin{pmatrix} 2 & i \\ -i & 3 \end{pmatrix}$ positive definite?

Check $G = G^H$ : Yes (conjugate transpose equals itself).

Eigenvalues: $\lambda = \frac{5 \pm \sqrt{25-20}}{2} = \frac{5 \pm \sqrt{5}}{2} > 0$

Both positive, so $G$ defines a valid inner product.

Theorem 7.14a: Cholesky Decomposition

A matrix $G$ is positive definite if and only if it can be written as:

G = L L^T

where $L$ is lower triangular with positive diagonal entries. This is the Cholesky decomposition.

Example 7.17a: Cholesky Decomposition Example

Factor $G = \begin{pmatrix} 4 & 2 \\ 2 & 5 \end{pmatrix}$ :

L = \begin{pmatrix} 2 & 0 \\ 1 & 2 \end{pmatrix}, \quad L L^T = \begin{pmatrix} 4 & 2 \\ 2 & 5 \end{pmatrix} = G

The positive diagonal in $L$ confirms $G$ is positive definite.

Theorem 7.14b: Sylvester's Criterion

A symmetric matrix $G$ is positive definite iff all leading principal minors are positive:

\det(G_1) > 0, \quad \det(G_2) > 0, \quad \ldots, \quad \det(G_n) > 0

where $G_k$ is the upper-left $k \times k$ submatrix.

Example 7.17b: Using Sylvester's Criterion

Is $G = \begin{pmatrix} 3 & 1 \\ 1 & 2 \end{pmatrix}$ positive definite?

$\det(G_1) = 3 > 0$ ✓
$\det(G_2) = 6 - 1 = 5 > 0$ ✓

Yes, $G$ is positive definite.

Remark 7.9a: Change of Basis

If we change basis with invertible matrix $P$ (new coordinates $y = Px$ ), the Gram matrix transforms as:

G' = P^T G P

Inner product is preserved: $y_1^T G' y_2 = x_1^T G x_2$ .

Theorem 7.14c: Diagonalization of Gram Matrix

Every real symmetric positive definite $G$ can be diagonalized by an orthogonal matrix $Q$ :

G = Q \Lambda Q^T

where $\Lambda = \text{diag}(\lambda_1, \ldots, \lambda_n)$ with $\lambda_i > 0$ .

9. Common Mistakes

Forgetting conjugation in complex inner products

For $\mathbb{C}^n$ , use $\langle x, y \rangle = \sum x_i \bar{y_i}$ , NOT $\sum x_i y_i$ . Without conjugation, $\langle x, x \rangle$ can be negative!

Assuming all norms come from inner products

Only norms satisfying the parallelogram law are induced by inner products. The 1-norm and ∞-norm are NOT inner product norms.

Confusing positive definite with positive semidefinite

Positive definite: $\langle x, x \rangle > 0$ for $x \neq 0$ . Positive semidefinite allows $\langle x, x \rangle = 0$ for some $x \neq 0$ . Only positive definite forms are inner products.

Wrong linearity convention

Some texts use linearity in the second argument. Be consistent! We use linearity in the first argument (physicist's convention).

Misapplying Cauchy-Schwarz

The inequality is $|\langle x, y \rangle| \leq \|x\| \cdot \|y\|$ , NOT $\langle x, y \rangle \leq \|x\| \cdot \|y\|$ . Don't forget the absolute value!

Confusing inner product with norm

The inner product $\langle x, y \rangle$ takes two arguments and can be negative. The norm $\|x\|$ takes one argument and is always non-negative.

Forgetting to check positive definiteness

When defining a new "inner product," always verify all three axioms. Positive definiteness is often the hardest to check—make sure $\langle x, x \rangle = 0$ only when $x = 0$ .

Assuming orthogonality is transitive

If $x \perp y$ and $y \perp z$ , it does NOT follow that $x \perp z$ . Example: $(1,0) \perp (0,1)$ and $(0,1) \perp (1,0)$ , but $(1,0) \not\perp (1,0)$ .

Mixing up triangle inequality directions

Triangle inequality: $\|x + y\| \leq \|x\| + \|y\|$ . Reverse triangle: $|\|x\| - \|y\|| \leq \|x - y\|$ . Don't confuse them!

10. Applications

Quantum Mechanics

Quantum states live in complex Hilbert spaces. The inner product $\langle \psi | \phi \rangle$ gives probability amplitudes. Orthogonal states are distinguishable.

Signal Processing

The $L^2$ inner product measures signal correlation. Orthogonal signals don't interfere. Fourier analysis uses orthogonal basis functions.

Statistics & Machine Learning

Covariance is an inner product. Correlation = cosine of angle. Kernel methods use inner products in feature spaces.

Computer Graphics

Lighting calculations use dot products (inner products). Surface normals and view directions determine shading.

Numerical Analysis

Least squares uses orthogonal projections. Krylov methods (like conjugate gradient) exploit inner products for efficient solving.

Approximation Theory

Best approximations minimize distance (norm). Orthogonal polynomials (Legendre, Chebyshev) arise from different inner products.

Remark 7.10a: Application: Least Squares

Given an inconsistent system $Ax = b$ , least squares finds $\hat{x}$ minimizing $\|Ax - b\|^2$ . The solution satisfies the normal equations:

A^T A \hat{x} = A^T b

This is the orthogonal projection of $b$ onto the column space of $A$ .

Example 7.18: Least Squares Fit

Fit a line $y = ax + b$ to points $(0, 1), (1, 2), (2, 2)$ :

A = \begin{pmatrix} 0 & 1 \\ 1 & 1 \\ 2 & 1 \end{pmatrix}, \quad b = \begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}

Normal equations: $A^T A = \begin{pmatrix} 5 & 3 \\ 3 & 3 \end{pmatrix}$ , $A^T b = \begin{pmatrix} 6 \\ 5 \end{pmatrix}$

Solving: $a = 1/2, b = 7/6$ , so the best-fit line is $y = \frac{1}{2}x + \frac{7}{6}$ .

Remark 7.10b: Application: Fourier Series

Any periodic function $f \in L^2[-\pi, \pi]$ can be written as:

f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos(nx) + b_n \sin(nx))

The coefficients are inner products: $a_n = \frac{1}{\pi}\langle f, \cos(nx) \rangle$ .

Remark 7.10c: Application: Data Science

In machine learning and data science:

Cosine similarity: Measures similarity between documents/vectors
PCA: Finds orthogonal directions of maximum variance
Kernel methods: Inner products in high-dimensional feature spaces

11. Key Takeaways

Inner Product Axioms

• Linearity in first argument
• Conjugate symmetry
• Positive definiteness

Induced Norm

$\|x\| = \sqrt{\langle x, x \rangle}$

Measures "length" of vectors

Cauchy-Schwarz

$|\langle x, y \rangle| \leq \|x\| \cdot \|y\|$

Most important inequality!

Orthogonality

$x \perp y \iff \langle x, y \rangle = 0$

Generalizes perpendicularity

Chapter Summary

Inner products generalize the familiar dot product to abstract vector spaces, providing the geometric concepts of length, angle, and orthogonality. The Cauchy-Schwarz inequality is the cornerstone result that makes everything work.

20+

Theorems

25+

Examples

12

Quiz Questions

10

FAQs

Essential Formulas to Remember

Core Definitions

• Inner product axioms: linearity, symmetry, positive definiteness
• Induced norm: $\|x\| = \sqrt{\langle x, x \rangle}$
• Orthogonality: $x \perp y \iff \langle x, y \rangle = 0$

Key Inequalities

• Cauchy-Schwarz: $|\langle x, y \rangle| \leq \|x\| \|y\|$
• Triangle: $\|x + y\| \leq \|x\| + \|y\|$
• Parallelogram: $\|x+y\|^2 + \|x-y\|^2 = 2\|x\|^2 + 2\|y\|^2$

Connections to Other Topics

Linear Maps

Adjoint operators, unitary/orthogonal matrices

Eigenvalues

Spectral theorem, orthogonal diagonalization

Applications

SVD, least squares, Fourier analysis

12. What's Next?

With inner products mastered, you're ready for:

Orthogonality (LA-7.2): Orthonormal bases, orthogonal sets, and their special properties
Gram-Schmidt (LA-7.3): The algorithm to construct orthonormal bases from any basis
Orthogonal Projections (LA-7.4): Best approximations and least squares
Spectral Theorem (LA-7.5): Diagonalization of self-adjoint operators

Study Tips for This Chapter

Practice computing inner products in different spaces (ℝⁿ, ℂⁿ, function spaces)
Memorize the Cauchy-Schwarz inequality and its proof—it's fundamental
Always check all three axioms when verifying an inner product
Draw pictures! Orthogonality, projections, and angles have geometric meaning
Connect back to familiar dot product intuition from ℝ²/ℝ³

Practice Problems to Try

Verify that the weighted inner product $\langle x, y \rangle_w = \sum w_i x_i y_i$ satisfies all axioms
Prove Cauchy-Schwarz using the quadratic discriminant method
Show that $\{1, x - \frac{1}{2}\}$ is orthogonal on $[0, 1]$
Find the orthogonal complement of $\text{span}\{(1, 1, 1)\}$ in $\mathbb{R}^3$
Prove that the parallelogram law fails for the 1-norm

Quick Reference

Key Formulas

• Norm: $\|x\| = \sqrt{\langle x, x \rangle}$
• Angle: $\cos\theta = \frac{\langle x, y \rangle}{\|x\|\|y\|}$
• Cauchy-Schwarz: $|\langle x,y\rangle| \leq \|x\|\|y\|$
• Pythagorean: $x \perp y \Rightarrow \|x+y\|^2 = \|x\|^2 + \|y\|^2$
• Parallelogram: $\|x+y\|^2 + \|x-y\|^2 = 2\|x\|^2 + 2\|y\|^2$
• Polarization (real): $\langle x, y \rangle = \frac{1}{4}(\|x+y\|^2 - \|x-y\|^2)$

Standard Inner Products

• $\mathbb{R}^n$ : $\langle x, y \rangle = x^T y = \sum x_i y_i$
• $\mathbb{C}^n$ : $\langle x, y \rangle = x^H y = \sum x_i \bar{y}_i$
• $L^2[a,b]$ : $\langle f, g \rangle = \int_a^b f(t)\overline{g(t)}\, dt$
• Matrices: $\langle A, B \rangle = \text{tr}(A^H B)$
• Weighted: $\langle x, y \rangle_w = \sum w_i x_i y_i$
• General: $\langle x, y \rangle_G = x^H G y$ (G pos. def.)

Inner Product Axioms

• Linearity: $\langle \alpha x + \beta y, z \rangle = \alpha \langle x, z \rangle + \beta \langle y, z \rangle$
• Conj. Symmetry: $\langle y, x \rangle = \overline{\langle x, y \rangle}$
• Pos. Definite: $\langle x, x \rangle \geq 0$ , $= 0 \iff x = 0$

Important Definitions

• Orthogonal: $x \perp y \iff \langle x, y \rangle = 0$
• Unit vector: $\|x\| = 1$
• Orth. complement: $S^\perp = \{v : \langle v, s \rangle = 0 \; \forall s \in S\}$

Inner Product Definition Practice

12

Questions

0

Correct

0%

Accuracy

1

An inner product

\langle \cdot, \cdot \rangle

must satisfy:

Easy

Not attempted

2

For the standard inner product on

\mathbb{R}^n

,

\langle x, y \rangle

equals:

Easy

Not attempted

3

The induced norm from inner product is:

Easy

Not attempted

4

Cauchy-Schwarz states that:

Medium

Not attempted

5

If

\langle x, y \rangle = 0

, we say

x

and

y

are:

Easy

Not attempted

6

For complex inner products,

\langle y, x \rangle

equals:

Medium

Not attempted

7

The parallelogram law states:

Medium

Not attempted

8

Which is NOT an inner product space?

Hard

Not attempted

9

The angle between vectors is defined as:

Medium

Not attempted

10

For

L^2[a,b]

, the inner product is:

Medium

Not attempted

11

Positive definiteness means:

Medium

Not attempted

12

The polarization identity relates:

Hard

Not attempted

Frequently Asked Questions

What's the difference between inner product and dot product?

The dot product is a specific inner product on ℝⁿ. An inner product is a more general concept that can be defined on any vector space satisfying the axioms (linearity, conjugate symmetry, positive definiteness). Every inner product induces a notion of 'dot product' in its space.

Why do complex inner products use conjugation?

Without conjugation, ⟨x,x⟩ could be negative or complex for nonzero x. Conjugate symmetry ensures ⟨x,x⟩ is always real, and positive definiteness makes it positive for x ≠ 0. This is essential for defining a valid norm.

What does positive definiteness guarantee?

It guarantees that ||x|| = √⟨x,x⟩ is a valid norm: (1) ||x|| ≥ 0, (2) ||x|| = 0 iff x = 0, (3) ||cx|| = |c|||x||. Without it, we couldn't measure 'length' properly.

How is Cauchy-Schwarz useful?

It's fundamental! It proves the triangle inequality, defines angles between vectors (since |cos θ| ≤ 1), bounds correlations in statistics, and appears throughout analysis, probability, and physics.

Can every norm come from an inner product?

No! A norm comes from an inner product if and only if it satisfies the parallelogram law: ||x+y||² + ||x-y||² = 2||x||² + 2||y||². The 1-norm and ∞-norm fail this test.

What's the significance of the polarization identity?

It shows that if you know the norm, you can recover the inner product (when it exists). This means all geometric information is encoded in lengths alone—angles are derived quantities.

Why study inner products beyond ℝⁿ?

Function spaces like L²[a,b] are infinite-dimensional inner product spaces crucial for Fourier analysis, quantum mechanics, and differential equations. The same geometric intuition (length, angle, orthogonality) extends to these spaces.

What's a Hilbert space?

A complete inner product space—meaning every Cauchy sequence converges. ℝⁿ and ℂⁿ are finite-dimensional Hilbert spaces. L²[a,b] is an infinite-dimensional example, fundamental in functional analysis.

How do inner products relate to matrices?

For finite-dimensional spaces, ⟨x,y⟩ = xᵀAy for some positive definite matrix A. The standard inner product uses A = I. Changing A changes the geometry (lengths and angles) of the space.

What's the difference between sesquilinear and bilinear?

Bilinear means linear in both arguments. Sesquilinear (Latin: 'one-and-a-half linear') means linear in one argument, conjugate-linear in the other. Complex inner products are sesquilinear; real inner products are bilinear.