So why isn't $\Bbb R^n = \oplus _{n = 1}^{m}\Bbb R^n$

Question

I thought that direct sum means each component of $V = \oplus U_i$ can be decomposed into elements of $U_i$. But if $U_i$ is replaced by the whole space, doesn't it mean the everything else in the direct sum doesn't matter?

Concrete case.

$\Bbb R = \Bbb R \oplus \Bbb R$, is this not true? All spaces are vector spaces over real numbers.

Answer 1

Yours is just a problem of definitions. Given two vector spaces $V$ and $W$ we have $$V+W=\{v+w|v\in V,w\in W\}$$ While the simbol $V\oplus W$ means that $V\cap W=\langle0\rangle$. So $$\mathbb{R}=\mathbb{R}+\mathbb{R}$$ and $$\mathbb{R}\neq\mathbb{R}\oplus\mathbb{R}$$ $$\mathbb{R}^2=\mathbb{R}\oplus\mathbb{R}$$.

Answer 2

I think you are confusing direct sum $\oplus$ and just the usual sum $+$ of subspaces. $\mathbb{R}\oplus \mathbb{R}$ is not isomorphic to $\mathbb{R}$ because they have different dimensions ($2$ and $1$ respectively). At the same time, $\mathbb{R}=\mathbb{R}+\mathbb{R}$ if you view $\mathbb{R}$ as a subspace of itself.