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326 norm space @{text E}, can be extended to a continuous linear form |
326 norm space @{text E}, can be extended to a continuous linear form |
327 @{text g} on @{text E} such that @{text "\<parallel>f\<parallel> = \<parallel>g\<parallel>"}. |
327 @{text g} on @{text E} such that @{text "\<parallel>f\<parallel> = \<parallel>g\<parallel>"}. |
328 *} |
328 *} |
329 |
329 |
330 theorem norm_HahnBanach: |
330 theorem norm_HahnBanach: |
331 includes normed_vectorspace E + subspace F E + linearform F f + fn_norm + continuous F norm f |
331 includes normed_vectorspace E + subspace F E + linearform F f + fn_norm + continuous F norm ("\<parallel>_\<parallel>") f |
332 shows "\<exists>g. linearform E g |
332 shows "\<exists>g. linearform E g |
333 \<and> continuous E norm g |
333 \<and> continuous E norm g |
334 \<and> (\<forall>x \<in> F. g x = f x) |
334 \<and> (\<forall>x \<in> F. g x = f x) |
335 \<and> \<parallel>g\<parallel>\<hyphen>E = \<parallel>f\<parallel>\<hyphen>F" |
335 \<and> \<parallel>g\<parallel>\<hyphen>E = \<parallel>f\<parallel>\<hyphen>F" |
336 proof - |
336 proof - |