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hope‧ful‧ly S1

1 [sentence adverb] a way of saying what you hope will happen, which some people think is incorrect :
Hopefully, I'll be back home by ten o'clock.
By then the problem will hopefully have been solved.
Hopefully, I'll be back home by ten o'clock.
By then the problem will hopefully have been solved.2 in a way that shows that you are hopeful :
'Will there be any food left over?' he asked hopefully.
'Will there be any food left over?' he asked hopefully.