Let $X$ be a compact Hausdorff space and let $K$ be a compact subset of $X.$ Let $$V = \left \{f \in C(X)\ \big |\ f(x) = 0,\ \text {for all}\ x \in K \right \}.$$ Then identify the quotient normed linear space $C(X)/V.$
For each $x \in K$ I define linear maps $T_x : C(X) \longrightarrow \Bbb C$ defined by $T_x (f) = f(x), f \in C(X).$ Clearly each $T_x$ is a bounded linear map with $\|T_x\|_{\text {op}} = 1,$ for all $x \in K.$ Hence $\text {Ker} (T_x)$ is a closed subspace of $C(X),$ for all $x \in K.$ So $V = \bigcap\limits_{x \in K} \text {Ker} (T_x)$ is also a closed subspace of $C(X).$ So $C(X)/V$ is a valid normed linear space endowed with the quotient norm. But I can't able think of a known normed linear space to which $C(X)/V$ is isomorphic. I am guessing that this quotient space has something to do with $C_c (X).$ But I am unable to figure out how? Any suggestion regarding this will be highly appreciated.
Thanks for your time.
Best Answer
$C(X)/V$ can be identified with $C(K)$. If $g \in C(K)$ there is an extension of $g$ to an element $G$ of $C(X)$ preserving the sup norm by Tietze Extension Theorem. If $H$ is another extension then $G-H$ vanishes on $K$ so $G+V=H+V$. Now look at the map $g \to G$. It is easy to check that this is injective. Given any element $f+V$ of $C(X)/V$ check that $f+V$ is the image of $g$ where $g$ is the restriction of $f$ to $K$. You can also verify that $\|G+V\|=\|g\|$ for all $g \in C(K)$.