Normed vector space

A normed vector space is a fancy way of talking about a place where we can do some math with lines and shapes.

Imagine you have a big piece of paper, and you want to draw a line on it. You can use a ruler to measure the length of the line, and you'll get some number that tells you how long it is. This number is called the "magnitude" of the line.

Now, imagine you have another line, and you want to compare its length to the first line. You can do this by comparing their magnitudes. If one line is longer than the other, then its magnitude will be larger.

A normed vector space is a place where we can do this kind of measurement, but with more complicated shapes than just lines. We can also add and subtract these shapes, and we can multiply them by numbers to make them bigger or smaller.

But there's one more thing that makes a vector space normed: we have a special way of measuring the magnitude of these shapes. We call this special way a "norm".

A norm tells us how big a shape is by giving us a number. This number represents the magnitude of the shape. And just like with lines, we can compare the magnitudes of different shapes to see which one is bigger.

So, to sum it up: a normed vector space is a place where we can do math with shapes, and we have a special way of measuring their magnitudes called a norm.