Hilbert Space and dual

Question

Let $X$ be a Hilbert Space and $\varphi \in X' \setminus \left\{0\right\}$. We write $$C=\left\{x \in X: \varphi(x)=1\right\}$$ and $$E=\left\{\lambda x: \lambda \in \mathbb{R}, x \in C\right\}.$$

Now I have to prove that, if $\varphi(x)=0$, then there exists a sequence $(x_h)$ in $E$, such that $$\lim_h x_h = x.$$ How can I find out this sequence? I thought that the statement is equivalent to the fact that $x$ is aderent to $E$, still I can't go ahead.

Thanks to those who can help me!

Tsemo Aristide · Accepted Answer · 2018-01-04T17:19:45.433

0

Since $\varphi$ is not trivial, there exists $u$ with $\varphi(u)=1$, consider $u_n=x+u/n$ where $n$ is a strictly positive integer, you have $\varphi(nu_n)=\varphi(nx+u)=1$, this implies that $u_n\in E$, $lim_nu_n=x$.

edited Jan 04 '18 at 17:19

answered Jan 04 '18 at 16:52

Tsemo Aristide

87,475

Do you mean there exists $u$ with $\varphi(u) \neq 0$, don't you? – Nicola M. Jan 04 '18 at 17:09
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Yes, I mean that – Tsemo Aristide Jan 04 '18 at 17:19
Thanks then! You helped me to solve it – Nicola M. Jan 04 '18 at 17:21

Hilbert Space and dual

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