Post History
#2: Post edited
What does it mean by saying that $C([0,1])$ is a subset of $L^\infty([0,1]))$?
- What does it mean by saying that $C([0,1])$ is a subset of $L^\infty([0,1])$?
>**Question.** What does it mean by saying that $C([0,1])$ is a subset of $L^\infty([0,1]))$?**Notes.** This is an example of questions that are quite "obvious" to experienced readers in analysis but may be very confusing to beginners who take the statement "literally": an element in $C([0,1])$ is a completely different object than an element in $L^\infty([0,1])$: the former is a function on $[0,1]$ while the latter is an _equivalence class_ of functions on $[0,1]$, how can one say $C([0,1])$ is a *subset* of $L^\infty([0,1]))$?- I will write my own answer below. Answers with other perspectives are all welcome.
- >**Question.** What does it mean by saying that $C([0,1])$ is a subset of $L^\infty([0,1])$?
- **Notes.** This is an example of questions that are quite "obvious" to experienced readers in analysis but may be very confusing to beginners who take the statement "literally": an element in $C([0,1])$ is a completely different object than an element in $L^\infty([0,1])$: the former is a function on $[0,1]$ while the latter is an _equivalence class_ of functions on $[0,1]$, how can one say $C([0,1])$ is a *subset* of $L^\infty([0,1])$?
- I will write my own answer below. Answers with other perspectives are all welcome.
#1: Initial revision
What does it mean by saying that $C([0,1])$ is a subset of $L^\infty([0,1]))$?
>**Question.** What does it mean by saying that $C([0,1])$ is a subset of $L^\infty([0,1]))$? **Notes.** This is an example of questions that are quite "obvious" to experienced readers in analysis but may be very confusing to beginners who take the statement "literally": an element in $C([0,1])$ is a completely different object than an element in $L^\infty([0,1])$: the former is a function on $[0,1]$ while the latter is an _equivalence class_ of functions on $[0,1]$, how can one say $C([0,1])$ is a *subset* of $L^\infty([0,1]))$? I will write my own answer below. Answers with other perspectives are all welcome.