The Manchester United fan waited patiently outside the Drake Stadium, where her team have spent the past two weeks training, holding a multicoloured placard stating: "Alexis Sanchez give me your shirt". There were no direct requests for items of clothing for any other United player from the hundred or so gathered fans.

The Chilean walked past and smiled, signed autographs and posed for selfies. He looked relaxed, but he needed his shirt to train under the scorching Los Angeles sun.

Though he'll be judged in far tougher games, Sanchez is impressing everyone with his appetite and enthusiasm to do well. This is nothing new -- if there was a criticism of him at Arsenal it was that he wanted to train too hard because he had a childlike enthusiasm for always wanting the ball at his feet -- but the form of Sanchez has been one of the bright spots of Manchester United's American preseason so far.

The Chilean missed a poor first game in Phoenix because of a visa issue, but brightened up an even worse second game in the high heat and humidity of Santa Clara.

Though Ander Herrera pipped him to the man-of-the-match award in Los Angeles on Wednesday, Sanchez was the standout. He looked better than everyone around him, just as he did on his United debut in the very contrasting surroundings of Yeovil in January, but that star quality failed to materialise for the rest of the season. Sanchez tried hard, but often cut a frustrated figure on the wing and admitted that he was exhausted. He is nothing if not honest. At former coub Barcelona, he admitted that he'd been "5/10" and didn't feel he deserved a place in the starting XI in 2013.

Like many of his teammates, Sanchez was awful against Tottenham at Wembley in the league but he headed an equaliser in the FA Cup semifinal against the same team at the same venue. Paul Pogba crossed for that goal, and it's those linkups between United's star men that fans want to see more of.

The pair were also on song in the second half of the 3-2 derby win at Manchester City. As he once showed with an audacious chip in a Camp Nou Clasico, Sanchez can be a star in the biggest games in football, but there were too few such moments in his first four months in Manchester. Three goals in 19 games wasn't impressive for a man who averaged 24 goals a season in his four full seasons before joining United. And, at 29, he won't be cut any slack for any inconsistencies of youth like Marcus Rashford or Anthony Martial.

Alexis Sanchez celebrates scoring in Manchester United's International Champions Cup win over AC Milan. John Peters/Man Utd via Getty Images

Sanchez's arrival at United disrupted the team too, but as Mourinho told ESPN: "It is not easy to come in the middle of the season. I'm not a big fan of the winter market because it is not easy for players to adapt. So I think after a few months with us, now to start the season I think it is a perfect moment for him to establish himself as an important player for us."

In Carson, California, Sanchez was deployed in a more advanced attacking role given the absence of Rashford, Martial and Romelu Lukaku. There was a school of thought when he was at Arsenal that he needed to be the main attacker to be at his best, and certainly Sanchez was rapid, lively, strong and daring in that role here. He scored the opening goal after 12 minutes and looked absolutely delighted after finishing a superb Juan Mata pass with confidence. A second-half pass for Ander Herrera showed that Sanchez can also create. Had the Spaniard scored, it could have saved a 26-goal shootout, but as Mourinho stated, the local fans enjoyed the drama and so did the players.

Sanchez played as the lead attacker in an unusual 3-4-2-1 formation that featured Andreas Pereira playing in the unusual position of a defensive midfielder. He, too, did well. Like Sanchez, he's technical, skilful and versatile, but the two are different players at very different stages.

Sanchez has been all smiles in the U.S., and not just because he's been meeting famous United fans including Gary Oldman and Julia Roberts.

As Ander Herrera told MUTV after the 9-8 penalty win against AC Milan: "Alexis told me that it was the first summer that he'd had some free time. I think we are going to see the best Alexis. He's really focussed and he's a fantastic professional and player. In attack we have a lot of options with dynamic, different players. I think we can be optimistic for this season."

Players are likely to say positive things about their teammates in public, but privately the word is that Sanchez is showing a determination that the last big-name South American attacker, Angel Di Maria, refused to show after things hadn't worked out for him in his first six months in Manchester.

Arsenal fans certainly didn't want to lose Sanchez, who was the man who sparked their team into life, a free-wheeling dynamo who scored important and often spectacular goals. They booed him like they did another Arsenal star who headed north, Robin van Persie. Sanchez felt he was joining a bigger club with a better chance of winning trophies. And a better pay packet, it must be said.

Sanchez is good enough to be the best player in the Premier League. It's vital that he shows his full potential this season, and he certainly looks as if he means business. Performing as well against Liverpool in Saturday's friendly in the 107,000-capacity Michigan Stadium will do him no harm either.