# Examples for $\|\lambda\|<\liminf_{n\to\infty}\|\lambda_n\|$ and examples for $\lim_{n\to\infty}\|\lambda_n\|$ does not exist

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?

• 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. – Michael Hardy Feb 18 '20 at 21:48

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 $$n$$.