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The full-wave rectified mains voltage is about √2 times the mains voltage, so about 170 V DC for 120 V RMS and about 340 V DC for 240 V RMS.
The CFL in your picture must be a 110 / 120 V version as the smoothing capacitor is rated to 300 V DC.
The DC is used to power the lamp via a current limiting circuit. That is basically a buck regulator, probably with a constant current output and something to provide filament current if the lamp doesn't strike.
All fluorescent lamps need a current limiting circuit. The old, heavy, CFLs and most larger fittings use an inductor directly connected to the mains. The mains frequency, voltage and the inductor value set the current. On those, the inductor is also used to generate the high voltage needed to strike a long tube.
A fluorescent lamp, regardless of shape, is an arc lamp. Works on the same principle as a neon lamp. The electrodes in a neon lamp are close together, the arc can start at a fairly low voltage. The electrodes of a fluorescent are quite a distance apart. They need a kicker to get them going. Such as the very old fixtures that used an external starter. Essentially a thermal timer. Modern technology has eliminated the need for such.
For all practical purposes, an arc is a short circuit. The ionized air around a DC contact is the conductor. Heavy DC contacts have "arc chutes" and "blow out coils" to extinguish arcs. AC usually will normally breaK its' own arc because the arc has to restart every time the line swaps polarity. Only in a very heavy current and short gap will AC maintain an arc. Compare AC to DC welders, for an example.
The gas in a neon or fluorescent lamp, once ionized, is the same. Neon gas for the "neon" lamp, mercury vapor in older fluorescent tubes. And most of the newer ones.
Once the arc is started, as a short circuit, it has no appreciable resistance to limit the current. Hence the series resistor in a neon lamp and the ballast in a fluorescent fixture. It's a current limiter. Same as the ballast in a mercury vapor or sodium HID fixture.
Neon gas fluoresces in the visible spectrum. A fluorescent in the ultra-violet region. A "black light" tube is merely a fluorescent lamp without the phosphors. (white powder) When the powder is struck by UV photons, it glows white in the visible spectrum. Different compositions here will create light of different colours. Hence the DL(daylight), CW(cool white), &c as part of the lamp number.
There are sea stories about playing with fluorescent lamps in the vicinity of high power transmitters. I never bothered to find out if the gas reacts to the RF or the RF caused a current internal to the lamp. Would be an interesting question for someone knowledgable in the physics of the matter.
If you will observe a neon lamp connected across DC, the glow is around the positive plate. On AC, it is around both, the AC swaps polarity (50)60 times a second. Each plate is positive for half of the period. A fluorescent is the same, although you don't see it. The phosphers will tend to light brighter on one end on DC. It was considered routine to swap the lamp end for end on a regular basis in a DC fixture. I don't know if modern ballasts eliminate the need for this or just no one bothers with maintenance anymore.
CFL's (compact fluorescent lamps) are still fluorescent lamps. The use of electronic ballasts has permitted generating curved arcs. But, the physics of the arc are no different from the 8 foot T-12 tubes in a commercial fixture. Merely lower power.
As an aside, it was not unusual to hear the old folks refer to a fluorescent light as a "neon" light fixture, supposedly after neon signs. Transformers / ballasts from neon signs are quite useful for serious high voltage tinkering.
So for clarification. The incoming AC is rectified and filtered to DC, then it goes through an inverter circuit and into a transformer. The reason for this is the inverter used in CFLS is very high frequency so the inductors can be quiet small compared to if you used 50-60hz mains where the coiles would be much larger. One of the things about the type of inverter used is they can without a load (before the arc is started) go up to quiet a high voltage, which starts the arc, once loaded the voltage drops immediatly down to whatever the coils current limit is. So by carefull coil choice you get your high starting voltage and current regulation in one simple circuit.
In the dark ages, (c.1966) small fluorescents (8-24 watts) were powered from DC with no oscillator. Fairly low voltage, too. 13 volts for RVs. The DC maintained the arc but did ruin one end of the tube. They needed to be rolled occasionally to equalize the "wear". The fixtures contained a ballast and a starter. The starter has been replaced with a more efficient ballast and the oscillator in more modern fixtures. I haven't pursued the changes in that area.
Granted, "Used to be" doesn't answer the question. But does give insight into the why of the answer. And I learned something here as well. Thanks....