Functions of Vectors
A function is a map \(f:\mathcal{V}\to\mathcal{S}\).
Easy to imagine two useful categories of such functions: field elements or vectors. You can map vectors to arbitrary sets.
Functional
A map from a vector to a field element is defined as a functional.
Operator
A map from a vector to a vector is defined as a operator.
Linear Maps
A map from any vector space \(\mathcal{V}\) is linear if it distributes over the vector space operations:
- \(f(\alpha|a\rangle + \beta|b\rangle) = \alpha f(|a\rangle) + \beta f(|b\rangle)\)
Linear Functionals - Dual Space
Definition
\(\mathcal{V}^*\) is the dual space of \(\mathcal{V}\), where elements of \(\mathcal{V}^*\) are all linear functionals \(f:\mathcal{V}\to\mathbb{F}\).
Notation: Define \(f(|a\rangle) \equiv \langle f|a\rangle\in\mathbb{F}\).
The Dual Space is a Vector Space.
Linear Operators
\(F(|a\rangle) = F|a\rangle=|Fa\rangle\)
Examples
- \(M: \mathbb{R}^n\to\mathbb{R}^m\). \(m\times n\) matrices.
- \(M: \mathcal{V}\to\mathcal{V}\), with \(\mathcal{V}\) being a funcation space. Derivatives, fourier transform, Laplace, integral.
Anti-Linear Maps
If \(\mathbb{F} = \mathbb{C}\) and \(f(\alpha|a\rangle + \beta |b\rangle) = \alpha^*f(|a\rangle) + \beta^* f(|b\rangle)\) then the operator is called anti-linear.