russel said:The exercise does not refer to a particular space. It is just a normed space X with an uncountable Hamel basis. A solution I came up with was to make a finite dimension closed subspace and show using baire that it is X, leading to a contadiction.
If the problem gave us a space for this example which one would that be?
A normed space is a mathematical structure that consists of a vector space and a norm, which is a function that assigns a non-negative length or size to each vector in the space. It is a generalization of the concept of length, which is commonly used in geometry and physics.
A Banach space is a complete normed space, meaning that every Cauchy sequence (a sequence where the terms get arbitrarily close to each other) in the space has a limit in the space. This ensures that the space is "filled in" and there are no missing points.
A normed space can fail to be Banach if there exist Cauchy sequences in the space that do not have a limit in the space. This can happen if the space is not "complete" in some sense, such as having missing points or not being large enough to contain all possible Cauchy sequences.
Yes, the space of continuous functions on a closed interval [a,b], denoted by C[a,b], is a normed space with the norm ||f|| = max{|f(x)| : x in [a,b]}. However, this space is not Banach because there exist Cauchy sequences of continuous functions that do not converge to a continuous function in the space. One such sequence is the sequence of polynomials that converge to the function f(x) = |x| on [-1,1].
Proving that a normed space is not Banach is important because it helps us understand the limitations of the space and how it differs from a complete space. It also allows us to identify the specific properties that are lacking in the space, which can then be studied and potentially remedied. Additionally, this proof may be necessary in order to apply certain theorems or techniques that require a Banach space.