OP is correct assuming "dual" is defined as all the linear functionals on a space. The dual of an infinite-dimensional space in this sense always has larger dim

OP is correct assuming "dual" is defined as all the linear functionals on a space. The dual of an infinite-dimensional space in this sense always has larger dimension than the original space. The difference comes in that for Hilbert spaces, "dual" is usually taken to mean only the continuous linear functionals.