Oh! My bad! I completely missed that the functions were continuous (it isn't required for π to be a metric)
OmnipotentEntity
joined 2 years ago
Careful β οΈ there is not guaranteed to be an element such that |π(x) - π(x)| is maximized. Consider π (x) = x if x < 3, 0 otherwise. Let π (x) = 0, and let the domain be [0, 4]. Clearly, the sup(|π (x) - π (x)| : x β [0, 4]) = 3, but there is no concrete value of x that will return this result. If you wish to demonstrate this in this manner, you will need to introduce an π > 0 and do some pedantic limit work.
Anyway, to prove this is a metric we must prove that it satisfies the 4 laws of metrics.
1. The distance from a point to itself is zero. π (π, π) = 0
This can be accomplished by simply observing that |π (x) - π (x)| = 0 βx β [a,b], so its sup = 0.
2. The distance between any two distinct points is non-negative.
If π β π, then βx β [a,b] such that π (x) β π (x). Thus for this point |π (x) - π (x)| > 0 and the sup > 0.
3. π (π, π) = π (π, π) β(π, π) in our space of functions.
Again, we must simply apply the definition of π observing that βx β [a,b] |π (x) - π (x)| = |π (x) - π (x)|, and the sup of two equal sets is equal.
4. Triangle inequality, for any triple of functions (π, π, π), π (π, π) + π (π, π) β₯ π (π, π)
For any (π, π, π) β βΒ³ it is well known that |π - π| β€ |π - π| + |π - π|, (triangle inequality of absolute values).
Further, for any two functions π, π we have sup({π (x) : x β [a, b]}) + sup({π (x) : x β [a, b]}) β₯ sup({π (x) + π (x) : x β [a, b]})
Letting π (x) = |π (x) - π (x)|, and π (x) = |π (x) - π (x)|, we have the following chain of implications:
π (π, π) + π (π, π) = sup({π (x) : x β [a, b]}) + sup({π (x) : x β [a, b]}) β₯ sup({π (x) + π (x) : x β [a, b]}) β₯ sup({|π (x) - π (x)| : x β [a, b]}) = π (π, π)
Taking the far left and far right side of this chain we have our triangles inequality that we seek.
Because π satisfies all four requirements it is a metric. QED.
QED stands for πΈβ‘π, naturally
Says the guy insinuating context to justify this attack, and accusing the woman attacked of lying, despite the fact that she was fired for not reporting the attack to a manager quickly enough?
At this point, you're just embarrassing yourself. Lemmy is still too small for everyone here not to see through this weak ass bait.
What game do you really want to play, but haven't yet because you feel it in your soul that it will get a remake/remaster soon enough?
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