There is no space, unless you left a vacuum, and sealed the container under this vacuum. And even then, the liquid will evaporate to fill the empty space with gas, up to some pressure (that depends on the composition of the liquid and temperature).
That said, if there are thus more than one phase in this container, what happens is dictated by Archimedes. When one phase is less dense than the other, then it will be sujected to a force equivalent to the "weight" or acceleration of its volume filled by the more dense phase. That, plus the surface tension of the interphase surface will dictate how bubbly the phases will change position during your shaking acceleration. This can lead to the creation of lots of small bubbles, up to mixing an "emulsion" with the right surface tension.
That said, if there are thus more than one phase in this container, what happens is dictated by Archimedes. When one phase is less dense than the other, then it will be sujected to a force equivalent to the "weight" or acceleration of its volume filled by the more dense phase. That, plus the surface tension of the interphase surface will dictate how bubbly the phases will change position during your shaking acceleration. This can lead to the creation of lots of small bubbles, up to mixing an "emulsion" with the right surface tension.