Barrelled space

Imagine you have a large toy box full of different kinds of toys. Now, imagine you only want to play with the toy cars in the box. So, you take all the other toys out and put them away. You now have a smaller box that only has toy cars in it, and this smaller box is called the barrelled space.

In math, we have something called a vector space, which is like a toy box full of different kinds of vectors (or objects). However, sometimes we only want to work with a specific kind of vector, just like how you only wanted to play with the toy cars. So, we take out all the other vectors in the vector space and only keep the ones that we want to work with. This new, smaller vector space that only has the specific vectors we want to work with is called the barrelled space.

The barrelled space has a special property where we can measure the size of the vectors inside it without using the exact distance formula that we use in the regular vector space. This makes it easier for us to do math with the vectors inside the barrelled space. It's like having a special tool to measure the toy cars in your toy box without having to use a ruler or a measuring tape.