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Hello i am currently struggling with a problem, F denotes either the set of all Real or Complex numbers.

The aim is to prove that ##F^\infty## is infinite dimensional.

First of i am not sure i understand the books definition of this collection. It says it consists of all sequences of elements of F.

My understanding is that it is simply ##\{x: x \in F^i, \lim_{i \to \infty}\} ## (Feels like there is a better way to write this) Maybe this way would be better. ##\{x: x \in (x_1, x_2,...), x_i \in F\} ##

if F where the natural numbers for simplicity would that be like ##\{(1,0,...), (1,1,0,...)...(2,0,0...), (2,2,0...),...(n1, n2, ...)\}##

Then to prove it, i am pretty sure doing it by proof contradiction would be good, by assuming it where finite dimensional, and hence it would have exist a basis for it. And then i am thinking maybe showing there is an element i can add to the basis, and it still would be a basis, hence contradicting the fact that all basis has the same size, or that any spanning set that are linear independent, becomes dependent if any element is added to that set.

I could be totally on the wrong track, feel i need some pointers on how to proceed.

Thank you

The aim is to prove that ##F^\infty## is infinite dimensional.

First of i am not sure i understand the books definition of this collection. It says it consists of all sequences of elements of F.

My understanding is that it is simply ##\{x: x \in F^i, \lim_{i \to \infty}\} ## (Feels like there is a better way to write this) Maybe this way would be better. ##\{x: x \in (x_1, x_2,...), x_i \in F\} ##

if F where the natural numbers for simplicity would that be like ##\{(1,0,...), (1,1,0,...)...(2,0,0...), (2,2,0...),...(n1, n2, ...)\}##

Then to prove it, i am pretty sure doing it by proof contradiction would be good, by assuming it where finite dimensional, and hence it would have exist a basis for it. And then i am thinking maybe showing there is an element i can add to the basis, and it still would be a basis, hence contradicting the fact that all basis has the same size, or that any spanning set that are linear independent, becomes dependent if any element is added to that set.

I could be totally on the wrong track, feel i need some pointers on how to proceed.

Thank you

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