The orthogonal of the orthogonal of a line?

Question

There is a theorem stating

$(A^\perp)^\perp=\overline {\text{span}(A)}$ where $\overline{\text{span}(A)}$ is the closure of the smallest linear space containing A.

Now, I was applying this to a very simple case.

Let's take a line $l$ orthogonal to a plane $P_1$, and let $P_2$ be an orthogonal plane to $P_1$ containing $l$.

We have that $l^\perp=P_1$ and $P_1^\perp= P_2$.

However, I don't see how $\overline{\text{span}(l)}=P_2$, since the span of $l$ is the line itself, and the closure of a line is again the line itself...

Where is my mistake?

https://math.stackexchange.com/questions/418271/orthogonal-complement-of-orthogonal-complement — An old man in the sea., Nov 18 '19 at 10:23

score 2 · Accepted Answer · answered Nov 18 '19 at 09:44

2

$P_1^{\perp}$ is not $P_2$. If you are working in $\mathbb R^{3}$ then $P_1^{\perp}$ is a line not a plane.

answered Nov 18 '19 at 09:44

Kavi Rama Murthy

359,332

thanks for the answer. could you give me a simple example of a non-closed subspace in R^3? – An old man in the sea. Nov 18 '19 at 09:56
2

In a finite dimensional normed linear space every subspace is closed. @Anoldmaninthesea. – Kavi Rama Murthy Nov 18 '19 at 10:03

score 1 · Answer 2 · answered Nov 18 '19 at 09:48

1

If you can find two planes that are orthogonal to each other, then the ambient space in which they live must have dimension at least $4$, and therefore $l^\perp =P_1$ is impossible, since $l^\perp $ must have dimension at least $3$.

answered Nov 18 '19 at 09:48

Arnaud Mortier

27,742

Arnaud, thanks for the answer, but I think you have a mistake. I agree with your first part. But the orthogonal of a line is in fact a plane in R^3... See the answer by Kabo – An old man in the sea. Nov 18 '19 at 09:58
@Anoldmaninthesea. As I explained, the fact that you have two orthogonal planes implies that the ambient space cannot be $\Bbb R^3$. – Arnaud Mortier Nov 18 '19 at 10:00
Ah... Ok, I get it now, thanks. By the way, could you give an example of a non-closed subspace in R^3? – An old man in the sea. Nov 18 '19 at 10:02
@Anoldmaninthesea. Kabo has answered that. – Arnaud Mortier Nov 18 '19 at 16:10

The orthogonal of the orthogonal of a line?

2 Answers2