Auxiliary normed space

Okay, so let's start with understanding what a normed space is. You know how we can measure the distance between two points using a ruler? When we talk about a normed space, we are basically talking about a space where we can measure distances between points using something called a "norm".

A norm is like a special type of ruler that is used to measure how long a vector (which is just a fancy way of saying an arrow) is. It's kind of like measuring the length of a piece of string - we use a ruler to do that. In the same way, a norm measures the length of a vector.

Now, coming to the topic of an "auxiliary normed space". An auxiliary normed space is just another normed space that is used to help us understand the properties of the original normed space better. It's like having a second ruler that we use to check if the first ruler is working correctly or not.

Think of it like this - imagine that you have a toy car and you want to make sure that its wheels are rotating properly. You could try turning the wheels by hand and seeing if they move smoothly. But it might be easier to use another toy car with good wheels and compare how the wheels in the two cars move. That second toy car is like the auxiliary normed space - it helps us compare and understand the original normed space better.

So, an auxiliary normed space is just a second normed space that helps us understand the first one better. It's a way to compare and analyze the properties of the original space.