Closed linear operator
A closed linear operator is like a magic machine that takes in things called "vectors" and does some fancy math to them. But unlike other machines, this one is special because it can handle lots and lots of vectors at the same time! We call this "linear" because it has some rules that make it easier for us to work with.
Now, sometimes when we stick a bunch of vectors into this magic machine, the answers we get out don't really make sense or they don't follow the rules we expect. That's why we need to "close" the machine, which means we only let certain types of inputs go in, and we only get certain types of answers out.
Think of it like making sure we only put the right puzzle pieces into the puzzle, and we get the right picture out at the end. By closing the operator, we make sure that it doesn't give us answers that don't fit our puzzle.
So, a closed linear operator is a magic machine that can handle lots of vectors at once, but we make sure it only gives us answers that make sense by limiting what we put into it and what we get out.
Now, sometimes when we stick a bunch of vectors into this magic machine, the answers we get out don't really make sense or they don't follow the rules we expect. That's why we need to "close" the machine, which means we only let certain types of inputs go in, and we only get certain types of answers out.
Think of it like making sure we only put the right puzzle pieces into the puzzle, and we get the right picture out at the end. By closing the operator, we make sure that it doesn't give us answers that don't fit our puzzle.
So, a closed linear operator is a magic machine that can handle lots of vectors at once, but we make sure it only gives us answers that make sense by limiting what we put into it and what we get out.