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We know that if $X$ is a normed space and $(\lambda_n)_{n\in\mathbb N}$ is a sequence in $X^*$ converging weak-* to $\lambda$, then $(\lambda_n)_{n\in\mathbb N}$ is bounded and $\|\lambda\|\le \liminf_{n\to\infty} \|\lambda_n\|$. This is almost immediate from the definition.

Is there any example for the equality fails? And is there any example for $\lambda_n\overset{*}\to\lambda$ but $\lim_{n\to\infty}\|\lambda_n\|$ does not exist?

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  • $\begingroup$ I see that Adrian Keister copy-edited the question. If the difference between $||a||$ and $\|a\|$ is too subtle for you, notice the difference between $||a|| ||b||$ and $\|a\|\|b\|.$ That is the reason why what Adrian Keister did is standard usage. $\endgroup$ – Michael Hardy Feb 18 '20 at 21:48
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Let $H$ be a Hilbert space with orthonormal basis $\{e_n\}$. The classic example of inequality is given by $\lambda_n \in H^*$ where $\lambda_n(x) = \langle x,e_n\rangle$. It is well-known that $\lambda_n \rightharpoonup 0$ yet $\|\lambda_n\| = 1$ for all $n$.

For the second question you can just do a simple modification of this, for instance $\lambda_n(x) = \langle x,e_n\rangle$ for even $n$ and $\lambda_n(x) = \langle 2x,e_n\rangle$ for odd $n$.

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