Inner product spaces

Overview §

An inner product is a function that takes two input vectors and outputs a scalar; this function must be positive, definite, linear on the first input, and have conjugate symmetry. The most important examples of inner products are the standard dot products on ${\mathbb{R}}^n$ and ${\mathbb{C}}^n$.

An inner product space is a vector space together with an inner product on it.

Related: Metrics, metric spaces, and the metric topology, Norms

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Real and complex dot products §

The real dot product of $x,y\in {\mathbb{R}}^n$ satisfies the following properties:

• Bilinear, or linear on inputs $x,y$ separately;
• Symmetric, or outputs the same value no matter the order of inputs;
• Has $x\cdot x=||x|{|}^2$, the length-squared of $x$;
• Has $x\cdot x=0⟺x=\vec{0}$.

Definition: Complex dot product

The complex dot product, or standard inner product on ${\mathbb{C}}^n$, of $z,w\in {\mathbb{C}}^n$ is defined by

$$\left(\begin{array}{c}{z}_1\\ \cdots \\ z_n\end{array}\right)\cdot \left(\begin{array}{c}{w}_1\\ \cdots \\ w_n\end{array}\right)=z_1\overline{w_1}+\cdots +z_n\overline{w_n}.$$

The complex dot product has the following properties:

• $z\cdot z=|z_1{|}^2+\cdots +|z_n{|}^2$, with $z\cdot z=0⟺z=\vec{0}$;
• Additive in both coordinates: $(z+z^{\prime })\cdot w=z\cdot w+z^{\prime }\cdot w$;
• Respects scaling in first coordinate only: $(cz)\cdot w=c(z\cdot w)$;
• Scaling the second coordinate is equal to scaling the dot product by the complex conjugate of that scalar: $z\cdot (cw)=\overline{c}(z\cdot w)$;
• Swapping the terms changes the dot product by a complex conjugation: $z\cdot w=\overline{w\cdot z}$.

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Inner product spaces §

Definition: Inner product space

An inner product space consists of two pieces of data:

• (D1) A vector space $V$ over the field $\mathbb{F}$, which is either $\mathbb{R}$ or $\mathbb{C}$;
• (D2) A function $⟨-,-⟩:V\times V\to \mathbb{F}$ which takes two input vectors and returns an element of the underlying field.

This data must satisfy four axioms:

• (I1) Positive. The quantity $⟨v,v⟩$, called the norm-squared, is a non-negative real number.
• (I2) Definite. $⟨v,v⟩=0⟺v=\vec{0}$.
• (I3) Linearity on the first input. For any $a,b\in \mathbb{F}$ and any $v,w,u\in V$, we have $⟨av+bw,u⟩=a⟨v,u⟩+b⟨w,u⟩$.
• (I4) Conjugate symmetry. For any $v,w\in V$, we have $⟨w,v⟩=\overline{⟨v,w⟩}$.
• (I3 + I4) Conjugate-linear, or antilinear, on the second input. $⟨v,aw+bu⟩=\overline{a}⟨v,w⟩+\overline{b}⟨v,u⟩$.

Note that subspaces of inner product spaces also have the properties of inner product spaces.

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Review §

• What are the five key properties of the real dot product? How do they differ from the complex dot product?