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If $X$ is a normed space and dim $x=\infty$, show that the dual space $X'$ (set of all bounded linear functionals on $X$) is not identical with algebraic dual space $X^*$ (set of all linear functionals on $X$)

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We have shown in

https://math.stackexchange.com/questions/1318751/prob-10-sec-2-10-in-erwin-kreyszigs-book

that there is at least one element $T\in X^*$ that is not in $X'$

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