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Yes. CO2 comes out of solution and occupies the head space in the container.
Then what occupies the space when it hasn't been shaken?
CO2, but at a lower pressure. Most of the CO2 is dissolved in the water.
Carbonated beverages are super saturated with respect to CO2. Meaning they force the water in the beverage to absorb more CO2 then it would like because they do it under pressure. This is why it bubbles off when you open the bottle and let it sit in the open air.
It only becomes supersaturated once you open it.
Saturation curves are scaled with respect to temperature and pressure. CO2 gas goes into solution in water at a rate proportional to the partial pressure of CO2 and inversely proportional to the temperature of the water.
In the beer industry, carbonation levels are measured in "volumes" of CO2. One volume is equivalent to one liter of gas at STP dissolved into one liter of beer. Atmospheric partial pressure can sustain just above 1 volume. So true cask beer, which some think of as uncarbonated, is actually just slightly carbonated. In order to fully degas any fermented beverage, you must boil it (used for testing purposes), run it through a pressure drop (filtration in the case of wine, for example), agitation or some other method.
Budweiser/Miller/Coors are very heavily carbonated, generally between 2.75 and 3 volumes. Most craft beer is 2.25-2.5 volumes or so. Belgians are generally higher that all of them - 3-3.5 volumes - as they are bottle-conditioned to a high degree. The American style brown 12oz. glass bottle is only rated to 3 volumes or around 12-14psig at room temperature. Belgians come in 11.2 ounce bottles because they have thicker walls. In order to fit in the same size 6/4-pack carriers and on shelves, the glass is thicker while the OD is the same - so, less volume.
Shaking a beer degasses it via agitation. This increases the partial pressure of CO2 inside the headspace until a new equilibrium point is reached. If allowed to sit and rest for a while, the gas will go back into solution until the original equilibrium point is restored.