In the beginning, during the alarm system test, the 'thief' walks into the book display room. As he walks by the officer at the door, the officer is standing with his hands clasped behind his back. The guard to the right of the display case (from this view, behind the 'thief') is also standing with his hands clasped behind his back. There is a cut to view the display case from the other side (in front of the 'thief'), and the officer beside the door, in the background, now has his arms crossed in front, and the guard to the left of the case (which was the guard on the right from the other view) now has his arms hanging at his sides. After the 'thief' takes the book, there is a cut back to the first angle, and the guard by the door has his hands clasped behind his back again. The guard to the right of the display case now has his hands clasped in front of his body.
During the explanation of the crime, Banacek causes the alarm to trip and the display room door to come crashing down. But the door bounces very slightly when it hits bottom, revealing that it is not a massive slab of steel but a much lighter prop.
Near the end when Banacek is describing how things went down he refers to an explosive called Gelignite. He mispronounces is as "jel-IG-nite." It's properly pronounced "JELL-ignite." Interestingly, in the Columbo season 4 entry "By Dawn's Early Light" Columbo makes the same mistake and is immediately corrected by Patrick McGoohan's character Col Rumford.
Update: During Banacek's initial investigation of the scene, he questions Elliot about the lock on the pedestal door, at which Elliot responds, "Blasted open with Gelignite." - also mispronouncing it as 'jel-IG-nite'. Banacek merely uses the same mispronunciation during his recounting of the events.
Update: During Banacek's initial investigation of the scene, he questions Elliot about the lock on the pedestal door, at which Elliot responds, "Blasted open with Gelignite." - also mispronouncing it as 'jel-IG-nite'. Banacek merely uses the same mispronunciation during his recounting of the events.
When the alarm/door system is being tested, Glassman speaks with the police, who indicate the power will be off in a few seconds. Glassman then arrives at the door and manually lifts it to enter the book display room. However, when the police arrive after the alarm is really set off, they had to have the door opened remotely.
Given the amount of technology Banacek has access to, he would have immediately recognized that the bundles of wire in the pedestal were all telephone wire (used by the telephone company to wire a house) - not something a manufacturer would use to build the alarm system.