Inner Product Space

Definition

An inner product on a vector space $V$ is an operation, $⟨ \cdot, \cdot ⟩ : V \times V \to R$ , on $V$ that assigns, to each pair of vectors $v, w$ a real number $⟨ v, w ⟩$ satsifying the follwoing conditions:

$⟨ v, v ⟩ \geq 0$
$⟨ v, v ⟩ = 0$ iff $| v ⟩ = 0$
$⟨ x, y ⟩ = ⟨ y, x ⟩$ for all $x, y \in V$
$⟨ α x + β y, z ⟩ = α ⟨ x, z ⟩ + β ⟨ y, z ⟩$ for all $x, y, z \in V$ and all scalars $α, β \in F$ .

An inner product space is defined as a vector space with an inner product: ( $V, (\cdot, \cdot)$ ).

Definition from class

Inner Product often it is useful to introduce a rule that maps pairs of vectors to numbers ( $C$ or $R$ ) with the properties:

$(| a ⟩, | b ⟩) = (| b ⟩, | a ⟩)^{*}$
The inner product is linear in the second argument, sesquilinear in the first argument
The only way to get zero with an inner product of a vector with itself is only if the vector is zero

Defintion - Orthogonality

$| a ⟩$ and $| b ⟩$ are orthogonal iff $(| a ⟩, | b ⟩) = 0$ .

Definition - Orthonormality

A basis is orthonormal if $(| i ⟩, | j ⟩) = δ_{j}^{i}$ .

Definition - 2-Norm

The inner product defines a norm on $V$ .

The 2-norm is given as $\sqrt{(| a ⟩, | a ⟩)} = \sqrt{⟨ a | a ⟩} = | | | a ⟩ | |$

Properties of IPS

For an orthonormal basis, $(| a ⟩, | b ⟩) = (\sum_{i} a^{i} | i ⟩, \sum_{j} b^{j} | j ⟩) = \sum_{i j} a_{i} b^{j} (| i ⟩, | j ⟩) = \sum_{i j} a_{i} b^{j} δ_{j}^{i} = \sum_{i} a_{i} b^{i} = \sum_{i} a^{i *} b^{i}$ .

Cauchy-Schwarz Inequality

$| (| v ⟩, | w ⟩) \leq | | | v ⟩ | | \cdot | | | w ⟩ | |$

Using Dual Basis

In an ON basis, $(| j ⟩, | a ⟩) = ⟨ j | a ⟩ = ⟨ j | \sum_{i} a^{i} | i ⟩ = a^{j}$ gets the components of $| a ⟩$ .

Similarly for $A_{j}^{i} = ⟨ i | A | j ⟩ = (| i ⟩, A | j ⟩)$

Relating the Dual Basis with the Inner Product

Suppose we have an inner product space $V$ . For any $| f ⟩ \in V$ we can always define the Functional $⟨ f |$ by $⟨ f | v ⟩ (| f ⟩, | v ⟩), \forall | v ⟩ \in V$ .

Thus, the existence of an inner product defines a natural map from the topological dual space and the vector space.

Reitz Representation Theorem

Classes of Operators

Aside

For any unitary operator $U$ [ $U^{†} = U^{- 1}$ ], there exists a Hermitian operator, $G$ [ $G^{†} = G$ ], for which $U = \exp (i G)$ . $G$ is called a generator of the Unitary symmetry.

Proof. Let $G$ be Hermitian. Then $\exp (i G)^{†} = \exp (- i G)$ . So, $\exp (- i G) \exp (i G) = \exp (i G) \exp (- i G) = \exp (0) = I$ . Hence $\exp (i G)$ is Unitary.

Outer Products

Projection Operators

Orthogonality of Operators

$Q^{T} Q = I$ .

Separable

Definition

An inner product space is separable if there exists a sequence of vectors ${| v_{i} ⟩}$ such that no (non-trivial) $| v ⟩ \in V$ is orthogonal to all the $| v_{i} ⟩$ . Such a set is called ``complete’’. An ON basis is a complete set of ON vectors iff $| v ⟩ = \sum_{i} | e_{i} ⟩ ⟨ e_{i} | v ⟩$ and $\sum_{i} | e_{i} ⟩ ⟨ e_{i} | = I$ .