# Geoemtric interpretation of the norm of a bounded linear functional

Printable View

• March 28th 2011, 11:38 PM
raed
Geoemtric interpretation of the norm of a bounded linear functional
Dear Colleagues,

Show that the norm $||f||$ of a bounded linear functional $f\neq 0$ on a normed space $X$ can be interpreted geometrically as the reciprocal of the distance $d=inf\{||x|| \ |f(x)=1\}$ of the hyperplane
$H_{1}=\{x\in X \ |f(x)=1\}$ from the origin.

Regards,

Raed.
• April 7th 2011, 06:19 PM
mr fantastic
Quote:

Originally Posted by raed
Dear Colleagues,

Show that the norm $||f||$ of a bounded linear functional $f\neq 0$ on a normed space $X$ can be interpreted geometrically as the reciprocal of the distance $d=inf\{||x|| \ |f(x)=1\}$ of the hyperplane
$H_{1}=\{x\in X \ |f(x)=1\}$ from the origin.
{"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 10, "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": 0.8736452460289001, "perplexity_flag": "head"}