The Hammers keeper is determined to leave Upton Park on a positive note prior to next summer's move to the Olympic Stadium.
The Hammers were handed a difficult start to the season, taking in trips to Arsenal, Liverpool and Manchester City in their first six games.
They have shocked the division with their start though, beating all three of those teams away from home as part of eye-catching performances - although their home form has been a disappointment.
Although they recently beat Newcastle United at Upton Park, both Leicester City and Bournemouth left east London with three points, and Adrian says they must focus on improving in front of their own supporters.
“When we all saw the calendar, we thought ‘what has befallen us!” he told Marca.
“After winning those games, now it seems easy. The important thing is to stay focused and improve at home because we lost two games, if we had scored, we would be even higher.”
The Champions League would be ideal
This season is an important one for the Hammers, with them set to leave their longstanding home of Upton Park for a move to the Olympic Stadium at the end of the season.
The squad are keen to enjoy a strong campaign so as to give the old ground a fitting send off, and their promising opening suggests that they are on the right track.
Adrian says everyone connected with the club is determined to leave on a positive note, with possible qualification for the Champions League a dream scenario.
“That would be ideal,” he added on the prospect of qualifying for the Champions League.
“Everyone is thinking about finishing the final season at Upton Park, before moving to the Olympic Stadium, with a good taste. It is early to think about that, but the score line is good, we must not slacken.”