---
tag: 11GMR0F2Q8
kind: definition
name: Rigged Hilbert space
msc: [46C99, 46A11, 81Q10]
defines: 6SPMVHRPM2
uses: [HM2HHE3J51, 4A82S5FVRT, XC6XPMC2FQ]
sources:
  - url: https://en.wikipedia.org/wiki/Rigged_Hilbert_space
---

## Definition

Let $H$ be a complex [[Hilbert space|HM2HHE3J51]] whose inner product $\ip{\cdot}{\cdot}$ is linear in the first argument.
Let $\Phi$ be a subspace of $H$ with a locally convex topology such that these conditions hold:

1. The inclusion $\Phi \to H$ is continuous.
2. $\Phi$ is dense in $H$.

Let $\Phi^\times$ be the space of continuous conjugate-linear maps $\Phi \to \C$.
Each $h \in H$ gives the element $\varphi \mapsto \ip{h}{\varphi}$ of $\Phi^\times$, and the resulting map $H \to \Phi^\times$ is linear and injective.
The chain $\Phi \subset H \subset \Phi^\times$ is a *rigged Hilbert space*, or *Gelfand triple*.
In the theory of generalized eigenvectors one also asks $\Phi$ to be a [[nuclear space|XC6XPMC2FQ]].

## Remarks

Injectivity of $H \to \Phi^\times$ comes from condition 2: if $\ip{h}{\varphi} = 0$ for all $\varphi$ in a dense subspace, then $h = 0$.
Many authors use the [[continuous dual|4A82S5FVRT]] $\Phi'$ of linear forms instead of $\Phi^\times$; then the map $H \to \Phi'$ is conjugate-linear.

Example: $\Phi = \mathcal{S}(\R)$, the Schwartz space with its Fréchet topology, $H = L^2(\R)$, and $\Phi^\times$ the conjugate-linear tempered distributions.
The inclusion is continuous because $\norm{\varphi}_{L^2} \le \sqrt{\pi}\, \sup_x \abs{(1 + x^2)^{1/2}\varphi(x)}$, and $\mathcal{S}(\R)$ is dense in $L^2(\R)$.
The delta $\varphi \mapsto \overline{\varphi(a)}$ lies in $\Phi^\times$ but not in $H$.

Non-example: let $\Phi$ be the Schwartz functions that vanish on $[0,1]$.
Its closure in $L^2(\R)$ consists of functions that vanish almost everywhere on $[0,1]$, so condition 2 fails.
Then every $h \in L^2(\R)$ supported in $[0,1]$ gives the zero functional on $\Phi$, and $H \to \Phi^\times$ is not injective.