src/HOL/Real/HahnBanach/HahnBanach.thy
changeset 16102 c5f6726d9bb1
parent 15206 09d78ec709c7
child 16417 9bc16273c2d4
equal deleted inserted replaced
16101:37471d84d353 16102:c5f6726d9bb1
   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 -