Inner product · aiwiki.page

An inner product is an operation on a real or complex vector space that assigns a scalar to each pair of vectors, subject to linearity, symmetry, and positivity conditions. It generalizes the familiar dot product of coordinate vectors, providing a way to define length, perpendicularity, and geometric relationships for objects such as functions and sequences. A vector space equipped with an inner product is called an inner product space. The concept connects linear algebra with geometry and analysis. (wiki.math.ntnu.no)

Definition and conventions

Let $V$ be a vector space over the field $\mathbb F$ , where $\mathbb F$ is either the real numbers or the complex numbers. Using the convention that the second argument is linear, an inner product is a map

\langle\cdot,\cdot\rangle:V\times V\longrightarrow\mathbb F

satisfying, for vectors $u,v,w$ and scalars $a,b$ :

Linearity in the second argument:
$\langle u,av+bw\rangle =a\langle u,v\rangle+b\langle u,w\rangle.$
Conjugate symmetry:
$\langle u,v\rangle=\overline{\langle v,u\rangle}.$
Positive definiteness: $\langle v,v\rangle$ is real and nonnegative, and equals zero exactly when $v=0$ .

These properties imply conjugate linearity in the first argument:

\langle au+bv,w\rangle =\overline a\langle u,w\rangle+\overline b\langle v,w\rangle.

Over the real numbers, conjugation has no effect, so the operation is symmetric and linear in both arguments. (ocw.mit.edu)

Many mathematical texts instead make the first argument linear and the second conjugate-linear. Both conventions describe the same structure, but formulas must be adjusted consistently. In complex spaces, an inner product is therefore sesquilinear, rather than bilinear. (linear.axler.net)

Coordinate examples

On $\mathbb R^n$ , the standard dot product is

\langle x,y\rangle=\sum_{j=1}^{n}x_jy_j.

On $\mathbb C^n$ , the corresponding formula under the second-argument-linear convention is

\langle x,y\rangle=\sum_{j=1}^{n}\overline{x_j}y_j.

Conjugation ensures that $\langle x,x\rangle=\sum_j|x_j|^2$ . Without conjugation, a nonzero complex vector can have zero or negative self-product, violating positive definiteness. (wiki.math.ntnu.no)

Inner products need not weight all coordinates equally. For a Hermitian positive-definite matrix $A$ ,

\langle x,y\rangle_A=x^*Ay,

where $x^*$ denotes conjugate transpose. Relative to a chosen basis, every finite-dimensional inner product has this form. Its representing matrix consists of the pairwise inner products of basis vectors, forming their Gram matrix. Changing the inner product can change lengths and perpendicularity without changing vector addition or scalar multiplication. (math.brown.edu)

Length, angles, and orthogonality

An inner product induces a norm:

\|v\|=\sqrt{\langle v,v\rangle}.

The associated distance is $d(u,v)=\|u-v\|$ , making the space a metric space as well as a normed vector space. The Cauchy–Schwarz inequality

|\langle u,v\rangle|\leq\|u\|\|v\|

controls the magnitude of the pairing. Equality holds exactly when the two vectors are linearly dependent. This inequality also yields the triangle inequality for the induced norm. (math.brown.edu)

For nonzero vectors in a real inner product space, their angle $\theta\in[0,\pi]$ is defined by

\cos\theta=\frac{\langle u,v\rangle}{\|u\|\|v\|}.

Vectors are orthogonal when their inner product is zero. Orthogonal vectors satisfy the generalized Pythagorean theorem:

\|u+v\|^2=\|u\|^2+\|v\|^2.

In complex spaces, inner products may be nonreal, so this real-angle formula does not apply unchanged. (math.brown.edu)

Orthonormal bases and projections

An orthonormal basis consists of mutually orthogonal unit vectors. The Gram–Schmidt process converts a finite linearly independent list into an orthonormal list with the same span by successively subtracting projections and normalizing. In an orthonormal basis, vector coordinates and lengths take particularly simple forms. (ocw.mit.edu)

If $e_1,\ldots,e_m$ form an orthonormal basis of a finite-dimensional subspace $W$ , its orthogonal projection is

P_Wv=\sum_{j=1}^{m}\langle e_j,v\rangle e_j.

The residual $v-P_Wv$ is orthogonal to every vector in $W$ , and $P_Wv$ is the unique element of $W$ minimizing distance to $v$ . This is the geometric basis of least-squares approximation, where a target vector is approximated within the column space of a matrix. (linear.axler.net)

Function spaces and completeness

For square-integrable complex functions on a measure space, an inner product is

\langle f,g\rangle=\int\overline{f(x)}g(x)\,d\mu(x).

The integral replaces the coordinate sum. Functions agreeing almost everywhere are identified, ensuring that zero norm means the zero element. The resulting $L^2$ space is a Hilbert space: an inner product space complete in its induced norm. Completeness means that every Cauchy sequence converges to an element of the space. Finite-dimensional inner product spaces are always complete; infinite-dimensional ones need not be. (ocw.mit.edu)

Orthogonal function expansions, including Fourier series, use inner products to extract coefficients. In quantum mechanics, complex inner products determine state overlaps. For normalized states, $|\langle\phi,\psi\rangle|^2$ gives the probability associated with projection onto $\phi$ , according to the Born rule. (ocw.mit.edu)

Characterization by the norm

Not every norm arises from an inner product. A real or complex norm is induced by an inner product exactly when it satisfies the parallelogram law:

\|u+v\|^2+\|u-v\|^2 =2\|u\|^2+2\|v\|^2.

The inducing inner product is unique. In the real case, the polarization identity recovers it as

\langle u,v\rangle =\frac{\|u+v\|^2-\|u-v\|^2}{4}.

Thus an inner-product norm encodes the entire pairing, not merely vector lengths. (linear.axler.net)