If we have a quotient space $E\backslash L_0$ where $E$ is a linear normed space and $L_0$ it's subspace the norm of an element $L$ in $E\backslash L_0$ is defined as $$\lVert L\rVert = \inf_{x \in L}{\lVert x\rVert}$$ What I'm trying to understand better is why we can find an element $x\in L$ s.t. $\lVert x \rVert < \lVert L \rVert + \epsilon$. I understand that such element is not going to be an infimum, but how are we sure that this element is going to be in $L$? Does is have something to do with the fact that $L$ is a closed set?
Thanks in advance!
Best Answer
This is immediate from definition of infimum. If this is not true then $\|x\| \geq \|L\|+\epsilon$ for all $x \in L$ which means $\|L\|+\epsilon$ is a lower bound; hence the greatest lower bound $\|L\|$ must be equal to or exceed this, i.e. $\|L\| \geq \|L\|+\epsilon$ which is a contardiction.