# Rudin's Functional Analysis Theorem 1.41

While I was reading the proof of Theorem 1.41 in Rudin's Functional Analysis, I was stuck in the equation $$\begin{equation*} \pi(\{x:d(x,0) where $$N$$ is a closed subspace of a topological vector space $$X$$, $$\pi$$ is the quotient map of $$X$$ onto $$X/N$$, $$d$$ is a complete translation-invariant metric on $$X$$, and $$\rho(\pi(x),\pi(y)) = \inf\{d(x-y,z):z\in N\}$$. Specifically, I cannot prove $$\{u:\rho(u,0). I guess that it seems necessary to show that for every $$x\in X$$ there exists $$y\in N$$ satisfying $$d(x,y) = \inf\{d(x,z):z\in N\}$$, but not sure. Would you give me any hint?

Fix $$u$$ with $$\rho(u,0), say $$u=\pi(y)$$. Since $$\inf \{d(y,z):z\in N\} there is some $$z\in N$$ with $$d(y,z). Then $$d(y-z,0) and $$\pi (y-z)=\pi(y)-\pi(z)=\pi(y)=u,$$ so taking $$x=y-z$$ does the trick.
• Why there is some $z\in N$ with $d(y,z)<r$? This is the point I cannot prove. Commented Apr 29, 2020 at 4:21
• @flyingwith Given any bounded non-empty set $A\subset \mathbb{R}$, if $\inf A <r$, then $r$ is not a lower bound of $A$ (because $\inf A$ is the largest lower bound), which means that there exists some $a\in A$ with $a<r$. Commented Apr 29, 2020 at 4:23
• Could you also say the infimum is attained in $N$ because it is closed? Commented Apr 29, 2020 at 4:36
• @Connor_Tracy No, it false for general Banach spaces (it holds in Hilbert spaces though, by taking orthogonal projections). Here's a counterexample: Let $X:=\{f\in C([0,1]): f(0)=0$. Let $\phi(f):=\int_0^1 f$ (wrt to Lebesgue measure), $N=\ker \phi$ (you can check that $\phi$ is continuous). Assuming you have a function $f\in X$ with $d(f, N)=1=\|f\|$, you can argue that $|\phi(f)|\|g\|\ge |\phi(g)|$ for every $g\in X$. If you consider $f_n(t)=t^\frac{1}{n}$, then $f_n\in X\setminus N$, and you can show using the previous fact that $|\phi(f)|\ge 1$. But $f\in M$, so $f(0)=0$. Commented Apr 29, 2020 at 4:45
• @Connor_Tracy Therefore, using that $\|f\|_\infty =1$, $\phi(f) \le \int _0^1 |f| <1$, contradiction. Commented Apr 29, 2020 at 4:51