# 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

1. $$M: \mathbb{R}^n\to\mathbb{R}^m$$. $$m\times n$$ matrices.
2. $$M: \mathcal{V}\to\mathcal{V}$$, 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.

Created: 2023-06-25 Sun 02:25

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