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A normed space $X$ is reflexive iff $X^{**}=\{g_x:x\in X\}$ where $g_x$ is bounded linear functional on $X^*$ defined by $g_x(f)=f(x)$ for any $f\in X^*$.

Let $X$ be a Hilbert space, would you help me to show that $X$ is reflexive.

One of the example is $L^2[a,b]$, the reason is its dual is $L^2$ and the second dual is $L^2$ again.

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    Your example for $X = L^2[a,b]$ can be extended to apply to any Hilbert space; this is the Riesz representation theorem.2012-12-11
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    @ChristopherA.Wong: Would u help me to check my answer below?2012-12-11

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