Infinite Dimensional Linear Algebra Proof

[*melinda*]

Prove that $\mathbf{F}^{\infty}$ is infinite dimensional.

$\mathbf{F}^{\infty}$ is the vector space consisting of all sequences of elements of $\mathbf{F}$, and $\mathbf{F}$ denotes the real or complex numbers.

I was thinking of showing that no list spans $\mathbf{F}^{\infty}$, which would mean that it had no basis and therefore could not be finite dimensional.

I'm just not sure if this approach works, and I'm also a little fuzzy on how to show this.

Another idea I had was to assume that $\mathbf{F}^{\infty}$was finite dimensional and show a contradiction, but again I am unsure of how to actually show this.

Any ideas or input would be much appreciated !

 

[StatusX]

The vectors (1,0,0,...), (0,1,0,...),... are all linearly independent. Does that help?

 

[matt grime]

I was thinking of showing that no list spans $\mathbf{F}^{\infty}$

You could show no *finite* list (or just set, unless list specifically means a finite set, but I've not come across that meaning before) is a spanning set.

which would mean that it had no basis

it definitely has basis, it is just not finite dimensional (in fact its dimension is the same as the cardinality of the real numbers).

I have nothing else to add other than follow the hint StatusX gave.

 

[*melinda*]

Thanks for all the help!

The book I'm working from does not discuss infinite dimensional vector spaces. It only gives a brief description of $\mathbf{F}^{\infty}$ and P(F), the set of all polynomials with coefficents in $\mathbf{F}$.

In particular it says, "because no list spans P(F), this vector space is infinite dimensional".

Does this mean that in infinite dimensions all you need for basis is linear independence?

 

[StatusX]

A basis is any linearly independent set of vectors that spans the space. There can be smaller lists that are linearly independent and larger lists that span the space, but the fact that a basis must have both properties forces it to have a unique cardinality that is characteristic of the space.

 

[matt grime]

A basis is a linearly independent spanning set. (Note there is no reference to cardinality at all.)

I have no idea why your book would choose the word list. Just think 'set'. No finite set of vectors can possibly span F[x] (which is what you ought to call the set of polynomials over F, not P(F) - I don't think I like this book, and I've not even seen it) or F^{oo}

 

[*melinda*]

Ironically, the book is called 'Linear Algebra Done Right' 2nd ed. by Sheldon Axler. I don't exactly love it, but it is what I'll be using this fall so I better get used to it.