What does it mean 'Infinite dimensional normed spaces'? I am currently taught about metric-spaces and Hilbert spaces and I got a bit confused, so I try to understand a simple concept to connect the pieces. 
Initially, I thought that a vectors space is a space which is defined by some axioms and the dimension of the vector space is defined by the number of basis (independet vectors). I assume then that a vector space of dimention $R^{n}$ is defined by n basis vectors and each vector in the space is $n$-dimensional. This is my understanding for vector spaces. 
When the normed spaces were introduced I think i missed something. Nomatter the norm used, the output of a norm is a scalar value. How can the normed-space then be infinite? 
Does it mean that it is a vector space of infinite dimensions (as I defined before), the norm of every vector satisifies the axioms of the vector space?
(ie if $x, y$ are in the vector space defined by the basis, then $norm(x)(x+y)$ is still in the same vector space?)
Thank you in advance
 A: Let $V$ be a vector space. A norm is a function $\|.\| : V\to [0,\infty)$. The norm has no bearing on whether $V$ is finite dimensional or infinite dimensional as it is a function from the vector space to the interval $[0,\infty)$.
You can check for yourself. The vector space $P(R)$ of all polynomials with real coefficients is infinite dimensional. But a norm function from $P(R)\to [0, \infty)$ makes perfect sense and has nothing to do with the finite or infinite nature of $P(R)$.
A: A normed space is a pair $(V,\|\cdot\|)$, where $V$ is a vector space and $\|\cdot\|$ is a norm over that vector space. The dimensionality of the space has nothing to do with the norm. For instance, the space $C^0[0,1]$ of continuous functions on $[0,1]$ is infinite dimensional, regardless of the fact that the norm
$$
\|f\|_0 = \max_{x\in[0,1]}|f(x)|
$$
yields only real values (as it is supposed to).
A: *

*A cleaner formulation of "dimension of the vector space is defined by the number of basis (independet vectors)" is:


The dimension of the vector space is defined as the cardinality of any basis, and the basis are the maximal subsets of independent vectors.

This is completely independent of any norm on the vector space, and the field of scalars in this definition is not necessarily $\Bbb R$ or $\Bbb C.$

*

*I don't know what you mean by "the norm of every vector satisifies the axioms of the vector space" but I agree with "if x, y are in the vector space defined by the basis, then norm(x)(x+y) is still in the same vector space" because (generally) when a vector space is equipped with a norm, the field of scalars is $\Bbb R$ (or $\Bbb C$) and the values of the norm are in $\Bbb R.$


*What holds most my attention is your "each vector in the space is $n$-dimensional". This is probably the central point of your confusion. In $V=\Bbb R^n,$ a vector is $n$-dimensional in the sense that it is defined by $n$ components, which are $n$ functions from $V$ to $\Bbb R.$ You seem bothered because a norm is also a function from $V$ to $\Bbb R,$ and is only one function. But that function does not define the vector, it is just an "additional information". And it is not linear, contrarily to the component functions.
