---
tag: NFN044536P
kind: concept
name: Krein space
msc: [46C20, 47B50]
sources:
  - url: https://en.wikipedia.org/wiki/Indefinite_inner_product_space
---

## Summary

A Krein space is a complex vector space with an inner product that may take negative values on some vectors.
It splits as an orthogonal sum of a positive part and a negative part.
The positive part is a Hilbert space for the form, and the negative part for its negative.
Krein spaces are the natural setting for operators that are self-adjoint for an indefinite form.
Such operators occur in relativistic quantum field theory.
When the negative part has finite dimension, the space is a Pontryagin space.

## Explanations

### L1 human

In ordinary geometry every nonzero vector has positive squared length.
In a Krein space some vectors have negative or zero "squared length", as in the space-time of special relativity.
The space still breaks into a part where lengths are positive and a part where they are negative, and each part looks like a Hilbert space.
The simplest example is $\C^2$ with $[x, y] = x_1 \overline{y_1} - x_2 \overline{y_2}$.