# Sheldon Axler's proof that every operator on a complex vector space has an eigenvalue

Since the proof is valid for any $$v \in V$$, the proof makes sure that any vector $$v$$ is an eigenvector which is of course not true. What is the error in this line of thought? Thanks a lot !

• You can get rid of the vector $v$ if you don't like it. Since $\mathcal L(V)$ is $n^2$-dimensional, $I,T,T^2,\ldots,T^{n^2}$ cannot be linearly independent and $p(T)=0$ for some nonzero polynomial $p$ of degree $m\le n^2$. This does not make $m\le n$ as in Axler's proof, but the rest of the argument remains essentially the same. – user1551 Jan 26 at 16:35
• The proof doesn't show that $v$ is an eigenvector... – David C. Ullrich Jan 26 at 19:43

We can have, for example, $$(T-\lambda_mI)v=u\neq0$$ and $$(T-\lambda_{m-1}I)u=0$$.
At the end of the proof it is only asserted that $$T-\lambda_i I$$ is not injective for some $$i$$. It does not give you $$(T-\lambda_i I)v=0$$ and we cannot say that $$v$$ is eigen vector corresponding to $$T-\lambda_i I$$.
The last statement does not mean that you can find such $$i$$ that $$(\hat{T}-\lambda_i \hat{I})\vec{v} = 0$$ for any $$\vec{v}$$. It rather means, that you can decompose you vector $$\vec{v} = a_1 \vec{u}_1 + \dots + a_m \vec{u}_m$$ so that every $$u_i$$ can find "its own" $$(\hat{T}-\lambda_i \hat{I})$$ and become zero.
{"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": 1, "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, "katex": 0, "math-container": 15, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.959516704082489, "perplexity": 102.55048933956179}, "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": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370506121.24/warc/CC-MAIN-20200401192839-20200401222839-00137.warc.gz"}