First, let's calculate how much Adrian spent on the poster boards. He bought 2 poster boards for $2.25 each:
\[ \text{Cost of poster boards} = 2 \times 2.25 = 4.50 \]
Next, we will determine how much money he has left after buying the poster boards:
\[ \text{Remaining amount} = 19.35 - 4.50 = 14.85 \]
Now, we know that the notebooks cost $1.65 each. To find out how many notebooks he bought, we divide the remaining amount by the cost of one notebook:
\[ \text{Number of notebooks} = \frac{14.85}{1.65} \]
Calculating this gives:
\[ \text{Number of notebooks} = \frac{14.85}{1.65} = 9 \]
Thus, the number of notebooks Adrian bought is
\[ \boxed{9} \]