# orthogonal complement - incorrect Brézis definition

I) Let $X$ be a Banach space. In his book Brézis introduces two definitions (section II.5):

1) If $M \subset X$ then he defines $$M^\perp=\{f \in X' \quad f(x)=0 \quad \forall x \in M\}$$ 2) If $N \subset X'$ then he defines $$N^\perp=\{x \in X \quad f(x)=0 \quad \forall f \in N\}$$ Isn't it absurd since, if $X$ is a Banach space then so is its dual $X'$ and therefore, for $N \subset X'$ the definition 1) gives $$N^\perp=\{f \in X'' \quad f(x)=0 \quad \forall x \in N\}$$ which does not correspond to the definition 2) unless $X$ is reflexive (which is not assumed there, this notion is introduced later in the book).

II) I also have another question. Assume now that $X$ is a Hilbert space. Suppose that, for whatever reason, I don't want to identify $X'$ with $X$ nor $X''$ with $X$. Then, how do you call the following subspace: $$\{x \in X \quad \langle x,m \rangle_H=0 \quad \forall m \in M\}$$ where $M$ is a subset of $X$ and $\langle x,m \rangle_H$ denotes the inner product of $X$ between the two elements $x,m \in X$.

The asymmetry in the definition is purposeful. The definition of $N^{\perp}$ requires that you know that you're working in the dual of a Banach space.This means that you will get different spaces if you consider $l^{\infty}$ as a Banach space by itself or as the dual of $l^1$. It will always be clear from context.