Banach–Mazur theorem

In functional analysis, a field of mathematics, a key object of study is a normed space, which is a vector space equipped with a norm, which allows vectors to be measured. When they are infinite dimensional, normed spaces can be very complicated. A standard normed space is the space ${\displaystyle C([0,1],\mathbb {R} )}$ of continuous functions on the unit interval, ${\displaystyle f:[0,1]\to \mathbb {R} }$, which is equipped with the norm ${\displaystyle \|f\|=\max _{x\in [0,1]}|f(x)|}$, the maximum value of the function.

The Banach–Mazur theorem is a theorem that provides one way of bounding the complexity of certain well-behaved normed spaces (separable). It states that every such normed space can be embedded into the normed space ${\displaystyle C([0,1],\mathbb {R} )}$, in such a way that the norm (length) of every vector is preserved (a property known as an isometry). It is named after Stefan Banach and Stanisław Mazur.