# Properties of adjoints of linear maps

I am studying linear algebra by myself, and I came across the need to prove the following, for adjoints of linear maps:

• ($f^\dagger)^\dagger=f$
• $(f\circ g)^\dagger=g^\dagger\circ f^\dagger$
• $\langle v|fw\rangle=\langle f^\dagger v|w\rangle$

Thank you very much!

• -
Do you know the definition of the adjoint? I strongly recommend that you do these yourself, because they're very easy. If you still can't do them, tell us what you have tried. – wj32 Oct 26 '12 at 12:59
Thank you. Off course, the first 2 are obvious if I accept the third. But if I don't, I don't know where to start. – dagger Oct 26 '12 at 15:42

1
The third is usually the definition of the adjoint. What is your definition? – wj32 Oct 26 '12 at 20:53

## 1 Answer

Meanwhile this was asked on MO and has an accepted answer: http://mathoverflow.net/questions/110739/

-
{"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": 3, "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": 0.9546091556549072, "perplexity_flag": "middle"}