The following is Corollary 4' in the book Functional Analysis, Peter D. Lax, chapter 8.
Every finite-dimensional subspace of $Y$ of a normed linear space $X$ has a closed complement.
I'm struggling to see where my counter-example is wrong.
Let $X$ be $\mathbb{R²}$ with the euclidean norm and $Y = \{ (x,x) | x \in \mathbb{R}\}$.
Then, according to above corollary, the set $Y^c
= \{ (x,y) \in \mathbb{R²}| x \ne y\}$ should be closed, right?
But it is not, since for example the sequence $x_n := (1, 1- \frac{1}{n}) \in Y^c$ does not converge in $Y^c$.
What am I missing?
Best Answer
There is a misunderstanding ! You have to show: if $\dim Y < \infty$, then there is a subspace $Z$ of $X$ such that $X= Y \oplus Z$ and $Z$ is closed.