The orthogonal of a function space.

Question

I have never heard the term orthogonal of a function space used before. After searching online I could not find much. Can anyone more knowledgeable explain this notion?

$\textbf{More information :}$

I am reading these notes http://www.cmap.polytechnique.fr/~allaire/homog/lect2.pdf where on page $10$ Lemma $1.1.8$ talks about the orthogonal of a function space. \

My (formal) guess :

Let $A$ be a function space and $B$ its orthogonal, then for all $f\in A,g \in B$ $$\int f\cdot g =0.$$

Answer 1

It's almost that. Given a subspace $A$ of $L^2\left(\Omega\times Y,H_\#^1(Y)^N\right)$, its orthogonal is the space$$A^\top=\left\{f\in L^2\left(\Omega\times Y,H_\#^1(Y)^N\right)\,\middle|\,(\forall g\in A):\int_{\Omega\times Y}fg=0\right\}.$$

Answer 2

As soon as you have an inner product $\langle \cdot, \cdot \rangle$, you can define the orthogonal of a linear subspace. This is exactly what is done here with the vector space $L^2$ equipped with the inner product $\langle f, g \rangle = \int f \cdot g$.

See Orthogonal complement for more details.