If $X$ is a normed space and $Y$ is a closed subspace of $X$. I need to prove that the space form by $span(\{x_{1},x_{2},...,x_{n}\}\cup Y)$ is also a closed subspace of $X$. The hint given by the lecturer is that we can make use of the quotient space $X/Y$. I don't have any background on topology so I have problems dealing with this.
Below are my questions:
- The quotient space $X/Y$ is the new space formed by identifying the elements in $Y$ am I right? Something like $D^{2}/S^{1}=S^{2}$.
- How does the quotient space $X/Y$ help in terms of proving that $span(\{x_{1},x_{2},...,x_{n}\}\cup Y)$ is closed?
- Do I have to prove by definition of closure that $span(\{x_{1},x_{2},...,x_{n}\}\cup Y)$ contains all the limits? Or is there an easier way?
Appreciate if anyone can explain to me how to approach this problem.
$X/Y$ is endowed with the structure of a normed space such that the quotient map $p:X\rightarrow X/Y$ is continuous. $V=span(p(x_1),...,p(x_n))$ is closed since it is finite dimensional this implies that $p^{-1}(V)$ is closed.