Inner product
High school
An inner product is a way of multiplying two vectors to get a number, generalizing the dot product, that gives a way to measure lengths and angles.
Undergraduate (real vector spaces)
Consider a real vector space $V$ and a function $\ip{\cdot}{\cdot} : V \times V \to \reals$. The function is an inner product if
- it is linear in its first argument: $\ip{a x + y}{z} = a \ip{x}{z} + \ip{y}{z}$ for all $x, y, z \in V$ and $a \in \reals$,
- it is symmetric: $\ip{x}{y} = \ip{y}{x}$ for all $x, y \in V$, and
- it is positive definite: $\ip{x}{x} > 0$ for for all $x \neq 0 in $V$.
Caveat. For complex vector spaces, symmetry is replaced by conjugate symmetry, $\ip{x}{y} = \overline{\ip{y}{x}}$, so the inner product is linear in one argument and conjugate-linear in the other.
Graduate (real or complex vector spaces)
Consider a vector space $V$ over $\mathbb{F}$, where $\mathbb{F}$ is $\reals$ or $\mathbb{C}$, and a sesquilinear form $\ip{\cdot}{\cdot}$ on $V$: a function $V \times V \to \mathbb{F}$ that is linear in its first argument and conjugate-linear in its second. The form is an inner product if, for all $x, y \in V$,
- it is Hermitian: $\ip{x}{y} = \overline{\ip{y}{x}}$, and
- it is positive definite: $\ip{x}{x} > 0$ for $x \neq 0$.
Over $\reals$, a sesquilinear form is a bilinear form, and Hermitian means symmetric.
Examples
Examples
-
$(x, y) \mapsto 2 x_1 y_1 + x_2 y_2$ on $\reals^2$
Undergraduate
$(x, y) \mapsto 2 x_1 y_1 + x_2 y_2$ is an inner product, because it is symmetric and positive definite.
-
$(x, y) \mapsto x_1 y_1 + x_2 y_2$ on $\reals^2$
Undergraduate
$(x, y) \mapsto x_1 y_1 + x_2 y_2$ is an inner product: the dot product on $\reals^2$.
Non-examples
Things that may be mistaken for examples.
-
$(x, y) \mapsto x y + 1$ on $\reals^2$
Undergraduate
$(x, y) \mapsto x y + 1$ is not a inner_product because it is not a bilinear form.
-
$x \mapsto \abs{x_1} + \abs{x_2}$ on $\reals^2$
Undergraduate
$x \mapsto \abs{x_1} + \abs{x_2}$ does not come from an inner product, because it fails the parallelogram law: with $x = (1, 0)$ and $y = (0, 1)$, $\norm{x + y}^2 + \norm{x - y}^2 = 8$, but $2 \norm{x}^2 + 2 \norm{y}^2 = 4$.
-
$(x, y) \mapsto x_1 y_1 - x_2 y_2$ on $\reals^2$
Undergraduate
$(x, y) \mapsto x_1 y_1 - x_2 y_2$ is not an inner product, because it is not positive definite: $(0, 1)$ paired with itself gives $-1$.
Distinctions involving this term
- Dot product vs. inner product vs. bilinear form: Bilinear forms, inner products, and the dot product are all ways of multiplying two vectors to get a number, from the most general to the most specific. The dot product is one particular inner product on $\reals^n$. An inner product is a bilinear form that is also symmetric and positive definite.
- Norm vs. metric vs. inner product: Norms, metrics, and inner products measure size, distance, and angle, and each one gives the next in a natural way. A norm measures the lengths of vectors, a metric measures distances between points, and an inner product measures both lengths and angles. An inner product gives a norm, and a norm gives a metric, but not the reverse.