---
tag: 0PXN9G4MX1
kind: statement
role: proposition
name: Orthogonal complement of the orthogonal complement
msc: [46C05]
uses: [079D68S3AW, 0E2NYPVSP9]
sources:
  - url: https://en.wikipedia.org/wiki/Orthogonal_complement
    locator: Finite dimensions
---

## Statement

Let $V$ be a finite-dimensional inner product space over $\R$ or $\C$, and let $W$ be a subspace of $V$.
Then
$$(W^\perp)^\perp = W,$$
where $W^\perp$ is the [[orthogonal complement|0XJXTBCB2T]] of $W$.

## Remarks

Each vector of the subspace is orthogonal to each vector of its orthogonal complement, so $W \subseteq (W^\perp)^\perp$. Both have dimension $\dim V - \dim W^\perp = \dim W$.
In an infinite-dimensional Hilbert space, $(W^\perp)^\perp$ is the closure of $W$.

## Explanations

### L1 human

Taking the orthogonal complement twice gives back the original subspace.
The complement of the complement of a line is the same line.