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#3: Post edited
by
Snoopy
·
2024-07-21T23:37:07Z (4 months ago)
improve wording
- >**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 question might seem "obvious" to those experienced in analysis but can confuse beginners who interpret the statement "literally": an element in \( C([0,1]) \) is fundamentally different from 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 $[0,1]$. Given this difference, how can we say \( C([0,1]) \) is a *subset* of \( L^\infty([0,1]) \)?
- I will provide my answer below. Answers offering different perspectives are all welcome.
#2: Post edited
by
Snoopy
·
2024-01-10T03:13:32Z (11 months ago)
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
by
Snoopy
·
2024-01-10T03:13:08Z (11 months ago)
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.