Symplectic vector space

A symplectic vector space is like a special type of playground where we can have a lot of fun with toys that move around in space. It's a like a big room with a special floor and walls that let us do cool tricks with our toys.

In our playground, we have two types of toys: vectors and forms. Vectors are like little arrows that show us where things are in space. Forms are like rules that tell us how to move our vectors around and have fun with them.

The special thing about our playground is that the rules for moving our toys are very special. They make sure that no matter how we move our toys around, certain things will always stay the same. For example, imagine that we have two vectors in our playground, call them A and B. Even if we move them around a lot, we will always be able to tell exactly how much they overlap (or don't overlap) with each other.

That's because the rules of our playground make sure that any time we move our toys around, we always preserve something called a "symplectic form". This is like a special set of rules that tells us how to play with our toys so that we always know exactly how much they overlap with each other.

Now, don't worry if this sounds a bit confusing. The important thing to remember is that in our special symplectic playground, we can have a lot of fun with our toys without ever losing track of how they relate to each other. It's a bit like playing a game of catch where no matter how far apart we stand or how many times we throw the ball, we always know where it's going to land. Fun, right?