Topological vector spaces

Okay kiddo, imagine you have a big toy box full of toys. You can play with them in many different ways, but they’re all just toys. Now, let's think about a big box of numbers instead of toys. Just like toys, numbers can be used in different ways, like adding, subtracting, multiplying, and dividing.

A topological vector space is like a big box of numbers, but with an extra set of rules. These rules say how we can put the numbers together and how they interact with each other. It's like a special way of playing with the numbers that makes them easier to understand and use.

The “topology” part of a topological vector space is like a set of instructions for how we should group the numbers. It helps us to understand how the different numbers relate to each other and what kind of “shape” they have when we look at them all together.

Imagine you have two toy boxes: one full of balls, and the other full of blocks. You can play with them separately and in different ways, but sometimes you can also mix them together to make a bigger toy. The same is true with topological vector spaces: we can combine different sets of “numbers” in different ways, depending on their properties and how they fit together.

So, topological vector spaces are like big boxes full of special numbers that have a set of rules telling us how to use them and combine them in different ways to understand their shape and properties.