Examples for the Kuratowski–Fréchet theorem

Question

I was wondering what kind of applications or basic exercises we can create with this powerful result https://en.wikipedia.org/wiki/Kuratowski_embedding

Thanks in advance!

Answer 1

I know at least one application of Kuratowski embedding which is the proof of the existence and unicity of the completion of a metric space $X$.

Take a metric space $(X,d)$. The map $\varphi : x \mapsto f_x$ from $X$ to $\mathscr C (X,\mathbb R)$ with $f_x(y)=d(x,y)-d(a,y)$ (for $a \in X$ fixed) is an isometry from $X$ to $\mathscr C (X,\mathbb R)$ by Kuratowski embedding theorem. The closure $\hat X$ of $\varphi(X)$ in $\mathscr C (X,\mathbb R)$ is a complete metric space and $X$ is isometric to $\varphi(X)$ which is dense in $\hat X$.

Answer 2

There is an application of the Kuratowski embedding in fixed point theory. First, let us give the definition of a hypercovex space:

Definition. A metrix space $M$ - equipped with a metric $d$ - is said to be hyperconvex if for any collection of points $\{x_i\}_{i\in\Gamma}$ in $M$, and real numbers $\{r_i\}_{i\in \Gamma}$, such that $d(x_{i}, x_{j})<r_i+r_j$, for any $i,j\in\Gamma$, we have $$ \bigcap_{i\in\Gamma} B(x_i, r_i) \neq \varnothing, $$ where $B(x,r)$ is the $d$-ball centered at $x$ with radius $r$.

The following result can be found in Chapter 4 (Fixed Point Theory in Hyperconvex Metric Spaces) of the book "Topics in Fixed Point Theory" (p.128):

Theorem. Let $M$ be a hyperconvex metric space and $T:M\to M$ a continuous mapping such that $\mathrm{cl}(T(B))$ is compact. Then $T$ has a fixed point.

To prove this fact we use the Kuratowski embedding $s$ of $M$ into $\ell_{\infty}(M)$. We then have the space $s(M)\subseteq \ell_{\infty}(M)$ whereon we define the mapping

$$ T_1:s(M)\ni y=s(x) \mapsto T_1(y) = s(T(x)) \in s(M). $$

Then $T$ has a fixed point if and only if $T_1$ has one. The rest of the proof is based on the fact that there is a non-expansive retraction from $\mathrm{co}(s(M))$ - the convex full of $s(M)$ - that is $R:\mathrm{co}(s(M))\to s(M)$, therefore, the mapping $T_1\circ R:\mathrm{co}(s(M)) \to \ell_{\infty}(M)$ is continuous. We can then prove that $T_1$ has a fixed point.

Another application of the Kuratowski embedding is that it allows us to properly define the addition of function $f,g:X\to Y$ (see J. Heinonen, Geometric embeddings of metric spaces, Lecture Notes, Univ. Jyvaskyla, 2003). If $Y$ is a metric space we can embed it isometrically into $\ell_\infty(Y)$ so we then can define

$$ f+g:X\to \ell_\infty(Y) $$