Writing the proof for the maximum norm
Clash Royale CLAN TAG#URR8PPP
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I am looking to prove that $|v | _infty = max_i |v_i|$ and show that it satisfies the norm conditions of:
- positivity
- scalar quantity
- triangle inequality
So far I know that:
$$| v |_infty = max _i | v_i |$$
Where $ v= (v_1, v_2, v_3, ldots v_n )$ and $n| v |_infty = nmax _i | v_i | = sum_i=1^n max_i |v_i|$
Then we say:
$$sum_i=1^n |v_i| leq sum_i=1^nmax _i |v_i| $$
Finally:
$$|v|_1 leq n|v|_infty $$
But, how can I confirm that the conditions are met?
matrices proof-verification norm
edited Sep 10 at 7:23
postmortes
1,58011016
asked Sep 10 at 6:59
frobeniusabnorms90776
1
add a comment |Â
up vote
-1
down vote
favorite
I am looking to prove that $|v | _infty = max_i |v_i|$ and show that it satisfies the norm conditions of:
- positivity
- scalar quantity
- triangle inequality
So far I know that:
$$| v |_infty = max _i | v_i |$$
Where $ v= (v_1, v_2, v_3, ldots v_n )$ and $n| v |_infty = nmax _i | v_i | = sum_i=1^n max_i |v_i|$
Then we say:
$$sum_i=1^n |v_i| leq sum_i=1^nmax _i |v_i| $$
Finally:
$$|v|_1 leq n|v|_infty $$
But, how can I confirm that the conditions are met?
matrices proof-verification norm
edited Sep 10 at 7:23
postmortes
1,58011016
asked Sep 10 at 6:59
frobeniusabnorms90776
1
1
If by positivity you mean norm nonnegative and 0 only when v=0, that is since norm is a max of nonnegative (because absolute values used), and only way to get 0 is all absolute values of coords are 0, in turn makes all coords zero.It's a scalar quantity since it's a number. Maybe yoou mean norm of kv is |k| times norm of v. The triangle inequality seems it wouldn't be hard, maybe try it first for only a few components of v and see what's going on.
– coffeemath
Sep 10 at 7:14
add a comment |Â
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
I am looking to prove that $|v | _infty = max_i |v_i|$ and show that it satisfies the norm conditions of:
- positivity
- scalar quantity
- triangle inequality
So far I know that:
$$| v |_infty = max _i | v_i |$$
Where $ v= (v_1, v_2, v_3, ldots v_n )$ and $n| v |_infty = nmax _i | v_i | = sum_i=1^n max_i |v_i|$
Then we say:
$$sum_i=1^n |v_i| leq sum_i=1^nmax _i |v_i| $$
Finally:
$$|v|_1 leq n|v|_infty $$
But, how can I confirm that the conditions are met?
matrices proof-verification norm
edited Sep 10 at 7:23
postmortes
1,58011016
asked Sep 10 at 6:59
frobeniusabnorms90776
1
I am looking to prove that $|v | _infty = max_i |v_i|$ and show that it satisfies the norm conditions of:
- positivity
- scalar quantity
- triangle inequality
So far I know that:
$$| v |_infty = max _i | v_i |$$
Where $ v= (v_1, v_2, v_3, ldots v_n )$ and $n| v |_infty = nmax _i | v_i | = sum_i=1^n max_i |v_i|$
Then we say:
$$sum_i=1^n |v_i| leq sum_i=1^nmax _i |v_i| $$
Finally:
$$|v|_1 leq n|v|_infty $$
But, how can I confirm that the conditions are met?
matrices proof-verification norm
matrices proof-verification norm
edited Sep 10 at 7:23
postmortes
1,58011016
asked Sep 10 at 6:59
frobeniusabnorms90776
1
edited Sep 10 at 7:23
postmortes
1,58011016
asked Sep 10 at 6:59
frobeniusabnorms90776
1
edited Sep 10 at 7:23
postmortes
1,58011016
edited Sep 10 at 7:23
postmortes
1,58011016
edited Sep 10 at 7:23
postmortes
1,58011016
1,58011016
asked Sep 10 at 6:59
frobeniusabnorms90776
1
asked Sep 10 at 6:59
frobeniusabnorms90776
1
asked Sep 10 at 6:59
frobeniusabnorms90776
1
1
1
If by positivity you mean norm nonnegative and 0 only when v=0, that is since norm is a max of nonnegative (because absolute values used), and only way to get 0 is all absolute values of coords are 0, in turn makes all coords zero.It's a scalar quantity since it's a number. Maybe yoou mean norm of kv is |k| times norm of v. The triangle inequality seems it wouldn't be hard, maybe try it first for only a few components of v and see what's going on.
– coffeemath
Sep 10 at 7:14
add a comment |Â
1
If by positivity you mean norm nonnegative and 0 only when v=0, that is since norm is a max of nonnegative (because absolute values used), and only way to get 0 is all absolute values of coords are 0, in turn makes all coords zero.It's a scalar quantity since it's a number. Maybe yoou mean norm of kv is |k| times norm of v. The triangle inequality seems it wouldn't be hard, maybe try it first for only a few components of v and see what's going on.
– coffeemath
Sep 10 at 7:14
1
1
If by positivity you mean norm nonnegative and 0 only when v=0, that is since norm is a max of nonnegative (because absolute values used), and only way to get 0 is all absolute values of coords are 0, in turn makes all coords zero.It's a scalar quantity since it's a number. Maybe yoou mean norm of kv is |k| times norm of v. The triangle inequality seems it wouldn't be hard, maybe try it first for only a few components of v and see what's going on.
– coffeemath
Sep 10 at 7:14
If by positivity you mean norm nonnegative and 0 only when v=0, that is since norm is a max of nonnegative (because absolute values used), and only way to get 0 is all absolute values of coords are 0, in turn makes all coords zero.It's a scalar quantity since it's a number. Maybe yoou mean norm of kv is |k| times norm of v. The triangle inequality seems it wouldn't be hard, maybe try it first for only a few components of v and see what's going on.
– coffeemath
Sep 10 at 7:14
add a comment |Â
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1
If by positivity you mean norm nonnegative and 0 only when v=0, that is since norm is a max of nonnegative (because absolute values used), and only way to get 0 is all absolute values of coords are 0, in turn makes all coords zero.It's a scalar quantity since it's a number. Maybe yoou mean norm of kv is |k| times norm of v. The triangle inequality seems it wouldn't be hard, maybe try it first for only a few components of v and see what's going on.
– coffeemath
Sep 10 at 7:14