Relations between annihilators and preannihilators in reflexive or Hilbert spaces.

Question

Let $X$ be a normed space, denote by $X^*$ the dual space $=$ the space of all continuous linear maps from $X$ to base field $\mathbb{K}$. Annihilator of $Y \subset X$ is $$ Y^{\bot} = \{f \in X^*:f(y) = 0 \;\; \forall y \in Y\}$$ Preannihilator of $Y \subset X^*$ is $$ {}^ {\bot}Y = \{x \in X: f(x) = 0 \;\; \forall f \in Y \}$$ Natural question is: How to describe this set's and relations between them in case of Hilbert space or reflexive space? For Hilbert space my thoughts are the following. Exists an canonical bijection $$R :H \longrightarrow H^*$$, and $R$ maps an orthogonal compliment of $Y$ onto annihilator of $Y$. Preannihilator of $Y$ is orthogonal compliment for the preimage of $Y$, so we know understand that for Hilbert space the notation given above is natural. For reflexive space a suppose that preannihilator and annihilator are just the same things, it's also easy to prove. By canonical embedding $$i: X \longrightarrow X^{**}$$we can think of $X$ and $X^{**}$ as the same spaces, and understanding that $$ {} ^{\bot}Y = Y^{\bot} \cap X$$ We get that for reflexive space it's just the same things. I have a strange feeling that i don't understand the connection with this things, am i right that: as every Hilbert space is reflexive, can we say the preannihilator of $Y$, annihilator $Y$, and orthogonal compliment of $Y$ are equal?