So the question asks:

> 
> Let $\Bbb R^{\Bbb R}$ be the vector space of all functions $f : \Bbb R\to \Bbb R$, and let $\Bbb R^{\Bbb N}$ be the vector space of all inﬁnite sequences, or tuples, $(x\_1, x\_2,\ldots , x\_n,\ldots)$ of real numbers.
> 
> 
> **(a)** Show that both $\Bbb R^{\Bbb R}$ and $\Bbb R^{\Bbb N}$ are inﬁnite-dimensional.
> 
> 
> **(b)** Prove that the function $T : \Bbb R^{\Bbb R}\to\Bbb R^{\Bbb N}$ given by $T(f) = (f(1), f(2),\ldots)$ is a linear transformation.Why doesn’t it make sense to talk about the matrix of the linear transformation T?
> 
> 
> 

So so far I have:

For **(a)** ,

Since the functions from $\Bbb R$ to $\Bbb R$ are polynomial, trigonometric, exponential and etc., by *Taylor series expansion*, every function has a polynomial representation of either finite degree or infinite degree.

So, the set of all functions over real numbers can be written as

$\Bbb R^{\Bbb R} = \{a\_0+ a\_1x + a\_2x\_2+\ldots+ a\_nx\_n +\ldots\mid a\_i\in\Bbb R\}$

So, each function can be written as n-tuples where n is finite or infinite as

$a\_0+ a\_1x + a\_2x\_2+\ldots+ a\_nx\_nn+\ldots=(a\_1, a\_2,\ldots, a\_n, \ldots)$

So, $\Bbb R^{\Bbb R}=\{(a\_1, a\_2,\ldots, a\_n,\ldots)\mid a\_i\in\Bbb R)\}$

This forms a vector space under addition of $n$- tuples and scalar multiplication.

Therefore, the function from $\Bbb R$ to $\Bbb R$ forms an infinite dimensional vector space $\Bbb R^{\Bbb R}$.

The space of all infinite sequences of real numbers can be written as

$\Bbb R^{\Bbb N}=\{(a\_1, a\_2,\ldots, a\_n,\ldots)\mid a\_i\in\Bbb R\}$.

Each sequence can be written as a linear combination of

$S = \{(1,0,0,\ldots,0,\ldots),(0,1,0,\ldots,0,\ldots),(0,0,1,0,\ldots,0,\ldots),\ldots\}$

So there are infinite vectors in $S$ which span $\Bbb R^{\Bbb N}$

Therefore, $\Bbb R^{\Bbb N}$ is an infinite dimensional vector space.

For **(b)**,

I know to prove a transformation is linear, it has to satisfy:

1. $T(v\_1+v\_2)=T(v\_1)+T(v\_2)\forall v\_1, v\_2\in V$, and
2. $T(kv)=kT(v)$ for any scalar $k$.

So, I need to prove that

1. $T(f\_1) +T(f\_2) = T(f\_1+f\_2)$
2. $T(k\cdot f\_1) = k T(f1)$

But, how am I supposed to do this? And the question asks why it makes no sense to talk about the matrix of $T$, is it because the transformation is from an infinite-dimensional vector space to the another infinite-dimensional vector space, and the matrix is abstract which has infinite basis?