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I have the question: "If $F_1$ and $F_2$ are subspaces of the hilbert space $H$, and they satisfy $F_1^\bot=F_2^\bot$, is it true that $F_1=F_2$?"

I would say that is it true, as one have that if $X$ is a linear subspace of a Hilbert space $H$, then $(X^\bot)^\bot=X$.

With the above fact I would show this with: $$F_1^\bot=F_2^\bot\quad\quad\Rightarrow\quad\quad (F_1^\bot)^\bot=(F_2^\bot)^\bot\quad\quad\Rightarrow\quad\quad F_1=F_2$$ However I doubt whether the argument is valid?

Bernard
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Frederik
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1 Answers1

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The argument is not valid (and the assertion is not true): The point is, that in general you only have $$ F^{\bot\bot} = \bar F $$ that is your argument only works for closed subspaces.

To give a counterexample, $F_1 = c_{00}$, $F_2 = c_0$ of $H = c_0$. Then $F_1^\bot = F_2^\bot = 0$, but $F_1 \ne F_2$.

martini
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