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He basically said Let (X, Σ, μ) be as above a probability space with a measure preserving transformation T, and let 1 ≤ p ≤ ∞. The conditional expectation with respect to the sub-σ-algebra ΣT of the T-invariant sets is a linear projector ET of norm 1 of the Banach space Lp(X, Σ, μ) onto its closed subspace Lp(X, ΣT, μ) The latter may also be characterized as the space of all T-invariant Lp-functions on X.
Makes perfect sense to me.
 
I love kicking your asses as pyro. First and foremost, make the attendant hat and forget stock. The degreaser is a must no matter what build you choose, and the outfit that it goes with is one of the few that gives a passive bonus, this one being extrarun speed. This is invaluable, especially as pyro.

Second, ignore the kids who say W+M1 is no skill. Quite the contrary, it is the secret to success. I know the maps well and basically run from one ammo box to the next, holding down the fire button. You lick invisible spies soooo frequently that it would be crazy to not spam the flames.

Third, memorize where your engineers have guns and make that part of your pathing. Engies are rooks and all the chessplayers reading know the power of two rooks in view of each other, game breaking power. If you save the guns once, you've made a major contribution to the round, and you will end up saving them many times if you are diligent and focused. Once the engie is up and running, other players will switch to engie because they realize spies aren't as much of a problem as usual. Don't bother with the homewrecker, you will be using the Powerjack, as it is part of the speed outfit.

Don't be scared of any class. They say heavies are your foil, but I burn the hell out of them same as anyone else. They can't keep a bead on you when you are in close, jumping around like a maniac, and will spray wildly as they burn, their vision further confused by the flames you pour onto them.

Program a button to do this: switc to melee from current weapon, then swing, then switch back to previous weapon. A button on the left side of my mouse does this, easy thumb access.

Experiment with the arc that flares and the detonator follows when you shoot. It's an amazingly easy shot to make, and extremely powerful because IF the enemy is on fire, it crits. Mad damage from the second shot on, or on a retreating burning enemy. Fuck the shotgun.

Finally, and this is important for the noob as well as the pro: learn to airblast. The timing is brutal at first, but oh my god if you and the airblast reach that point where you 'click', there is nothing like the awesome power that a good airblaster adds to the team. You neutralize ubers, reject demo spam, push melee classes like the backstabbing spy away from your back and head, send the soldier's missle back at him, you literally control the flow of the game. But that's advanced and very difficult.

Hope this helps. Also, anytime you get advice on a class and people tell you the same thing they red on a forum or on reddit or whatever, ignore them. If one more dork tells me Im noob for using the Black Box or the Powerjack or Kritz on offense or that I'm unskilled as all I do is W+M1, I will explode.
 
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I think the W+M1 complaint's been used up or something.

Some times you can go several rounds using exclusively W and M1. No one says a word about it.
 
I think the W+M1 complaint's been used up or something.

Some times you can go several rounds using exclusively W and M1. No one says a word about it.
Honestly when people complain about W+m1. I wonder if they ever played Pyro when TF2 first came out...
Personally I've seen a Phlog Pyro wipe an entire team and nobody complains about it.
 
I've seen a tonne of advice saying you should switch out the Flamethrower for the Degreaser, but i've recently been using the Stock Flamethrower / Flare Gun / Axtinguisher, and it's surprisingly easy to do well with, it's also extremely flexible. If you can aim well enough with the Flare, you're a little more useful mid-long range, unlike the Shotgun.
 
I've seen a tonne of advice saying you should switch out the Flamethrower for the Degreaser, but i've recently been using the Stock Flamethrower / Flare Gun / Axtinguisher, and it's surprisingly easy to do well with, it's also extremely flexible. If you can aim well enough with the Flare, you're a little more useful mid-long range, unlike the Shotgun.

I've been using that setup lately due to me playing pyro a little more to show off my new pyro robo hats. The reduced damage and afterburn from the Degreaser isn't worth the weapon switch speed, unless you are using the reserve shooter, too. Long distance crits against a flaming sniper are incredibly satisfying, as is a crit on a fleeing scout.