# Show $\{u_n\}$ orthonormal, A compact implies $\|Au_n\| \to 0$

I'm having a bit trouble with this homework exercise.

Let $\mathcal{H}$ be a Hilbert space and $\{u_n\}_{n=1}^\infty$ an orthonormal sequence in $\mathcal{H}$. Let $A$ be a compact operator on $\mathcal{H}$. Show that $\|Au_n \| \to 0$ as $n\to \infty$.

My book defines a compact operator as an operator $A$ such that whenever $f_n$ is bounded, then $Af_n$ has a convergent subsequence (equivalently, the image of $A$ is relatively compact).

It seems I must somehow combine the fact that $Au_n$ has a convergent subsequence with the fact that $\{u_n\}$ is orthonormal. This is where I get stuck. Maybe I can somehow use the fact that $\|u_n-u_m \| = \sqrt{2}\,$ for $m \neq n$.

-

If you know $u_n$ converges weakly, and find what its weak limit is, that may help you. – GEdgar Oct 16 '11 at 17:01

## 1 Answer

I suppose it is clear for you that a compact operator is clearly bounded so continuous. Suppose that the $(Au_n)$ has a convergent subsequence towards $v$ not zero. For the sake of simplicity, let us note also $(Au_n)$ this subsequence. $(u_n)$ is then also an orthonormal sequence in $\mathcal{H}$. Set $v_n=\frac{1}{n}\sum_{k=n}^{2n}u_k$. It is clear that the sequence $(v_n)$ converges towards $0$. But $(Av_n)$ converge towards $v$ which is not zero. QEA.

-

Thanks for the answer. However, doesn't this just show that the convergent subsequence of $Au_n$ converges to zero? It doesn't show that $Au_n$ itself converges. (I guess it now suffices to show, for example, that $Au_n$ is Cauchy) – Fredrik Meyer Oct 16 '11 at 17:13
@FredrikMeyer it shows actually that ANY convergent subsequence of $(Au_n)$ converges towards $0$. Therefore $(Au_n)$ converges as a whole towards $0$ (if not, being relatively compact as a set, you can easily extract a subsequence not converging towards $0$ ...) – brunoh Oct 16 '11 at 17:27
Of course! Thanks! – Fredrik Meyer Oct 16 '11 at 17:28
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 36, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9535212516784668, "perplexity_flag": "head"}