Infinite-dimensional normed space meaning

Question

I am reading about infinite-dimensional normed spaces but am struggling to understand the real meaning of it. Is there a simple explanation of this term?

Answer 1

"Dimension" in the context of vector spaces is usually defined to be the cardinality (or number, in finite-dimensional cases) of elements in a basis of said vector space. An infinite-dimensional vector space is simply one which has no finite basis.

See also this question.

Answer 2

For a vector space $V$ over the field $\mathbb{F}$ (this can be the reals or the complex numbers if you like) we define a basis to be a subset $B \subset V$ such that if $x \in V$ there exist a unique and finite number of elements $b_1, \ldots, b_n \in B$ and $\lambda_1, \ldots, \lambda_n \in \mathbb{F}$ such that $x = \sum_{i=1}^n \lambda_i b_i$. We say that $V$ is infinite dimensional if all such possible basis $B$ we have that $|B|$.

This is what is called a Hamel basis in the literature. You should be careful to not mix this up with Schauder basis which is often used for Banach spaces and Hilbert spaces (an orthonormal basis of a Hilbert space would be an example).