# Inner-product spaces

## Alexei Gilchrist

### 1 Inner products

An Inner-product, or *scalar-product*, is a map from a pair of vectors to a complex number. It can be represented by a simple tuple: \((\bullet,\bullet)\), and is defined for the entire vector space \(\mathbb{V}\):

The inner product helps capture the concepts of *angle* and *length* from spatial vectors, and will allow us to define orthogonality and then orthonormal bases.

An inner product has the following three properties:

- Linearity
- The inner-product is linear in the
*second*component:\[(|x\rangle,a|y\rangle+b|z\rangle) = a(|x\rangle, |y\rangle)+b(|x\rangle,|z\rangle)\] - Symmetry
- Swapping the order of the arguments produces the complex-conjugate of the result \[(|\psi\rangle, |\phi\rangle) = (|\phi\rangle,|\psi\rangle)^*\]
- Positivity
- An inner product of a vector with itself is always a non-negative real number. \[ (|\psi\rangle,|\psi\rangle) \ge 0 \]

Note that the linearity and symmetry properties imply the innner-product will be *anti-linear*, or *conjugate-linear* in the first component: \((a|x\rangle+b|y\rangle,|z\rangle) = a^*(|x\rangle, |z\rangle)+b^*(|y\rangle,|z\rangle)\). Be aware that some books use the convetion that the inner-product is linear in the first component but the reason for using the second component will become clear when we introduce the Dirac notation.

Say \( |a\rangle,|b\rangle \in \mathbb{C}^n \), i.e. the vector space is a stack of \(n\) complex numbers like \((a_1,a_2,\ldots,a_n)\). Then the inner product is defined as

Square integrable functions are also vectors. These are functions \(f(x)\) such that \(\int_{-\infty}^{\infty} |f(x)|^2 dx\) is finite. The inner-product for these functions is defined as

It’s a good exercise to verify that both examples above satisfy *linearity*, *symmetry*, and *positivity*.

Two vectors \(|a\rangle\), and \(|b\rangle\) are said to be **orthogonal** if their inner product is zero \((|a\rangle,|b\rangle)=0\).

### 2 Norm

A *norm*, denoted with double bars \(\lVert \bullet\rVert\), associates a positive real number to every vector

*length*of a vector. To do so it obeys three properties:

Given an inner-product, we can define a “natural” norm as

### 3 Basis

One of the key tools that we will use in quantum mechanics is that of a basis, and in particular orthonormal bases. So let’s start with some definitions.

#### 3.1 Definitions

*span*of \(s\) is the set of all vectors formed by linear combinations of the vectors in \(s\).

A *basis* is a set of vectors in \(\mathbb{V}\) that are linearly independent, and the span of the vectors is the whole vector space \(\mathbb{V}\).

They are useful because any vector \(|y\rangle\) can be expanded interms of the basis.