Sheriff's deputies also investigating at nearby Manzanita Elementary for a report of gunshots, but they reported the school is clear and have found no evidence of shots fired or injuries.
The sheriff's department said on its Twitter account that the incident was an "isolated shooting".
The police also stated that the suspect fled the scene but was arrested at a supermarket by an off-duty police officer.
The victim was expected to make a full recovery, he said. Police arrived three minutes later. "I heard a whole bunch going off". "They basically said, 'Mr".
A student's mother, Ana Salazar, told CNN affiliates KCAL and KCBS that her son texted her to say a shooting had happened.
"I thought he was joking, and I'm like, 'Don't mess around, that's not amusing, '" said mother Ana Salazar. I heard it at the other side of the school. "And I'm locked up in one of the buildings now'".
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"We are all concerned about the safety of our children", Palmdale School District Superintendent Raul Maldonado said in an alert.
"I want to let you know that all of the Palmdale Elementary Schools were put on lockdown, and will stay on lockdown until the situation is confirmed as safe for our students, teachers, staff and parents". Both were students at the school.
Palmdale is about 40 miles north of Los Angeles.
The male suspect was detained after reports of gunfire and calls saying an armed man had been spotted on campus around 7 a.m., an hour before the start of classes.
The initial reports of a possible school shooting drew immediate attention from major news outlets and cable TV networks.
The deadliest school shooting this year was on February 14, when a former student killed 17 students and staff at Marjory Stoneman Douglas High School in Parkland, Florida.