# Metric Spaces: Basic Concepts

Definition 1 A metric space is a pair $$(X, \rho)$$ of a set $$X$$ and a distance (a single-valued nonnegative real function) $$\rho$$ such that for all $$x, y \in X$$:

1. $$\rho(x, y) = 0$$ iff $$x = y$$;

2. $$\rho(x, y) = \rho(y, x)$$ [symmetry];

3. $$\rho(x, z) \leq \rho(x, y) + \rho(y, z)$$ [triangle inequality].

Examples of different metric spaces are:

1. discrete space;

2. $$R^1$$;

3. n-dimensional Euclidean space;

4. $$R_1^n$$;

5. $$R_0^n$$;

6. $$C_{[a, b]}$$;

7. $$l_2$$;

8. $$C_{[a, b]}^2$$;

9. $$m$$;

10. $$R_p^n$$;

11. $$l_p$$.

Consult the book for more detail.

Homogeneous if it holds for two points $$(a_1, \ldots, a_n)$$ and $$(b_1, \ldots, b_2)$$, then it holds for $$(\lambda a_1, \ldots, \lambda a_n)$$ and $$(\mu b_1, \ldots, \mu b_n)$$ where $$\lambda$$ and $$\mu$$ are arbitrary real numbers.

When showing the triangle inequality, several inequalitites are used and named.

Cauchy-Schwarz inequality: $(\Sigma_{k=1}^n a_k b_k)^2 \leq \Sigma_{k=1}^n {a_k}^2 \Sigma_{k=1}^n {b_k}^2$

Schwarz’s inequality: $(\int_a^b x(t) y(t) dt)^2 \leq \int_a^b x^2(t) dt \int_a^b y^2(t) dt$

Minkowski’s inequality: $(\Sigma_{k=1}^n |a_k + b_k|^p)^{\frac{1}{p}} \leq (\Sigma_{k=1}^n |a_k|^p)^{\frac{1}{p}} + (\Sigma_{k=1}^n |b_k|^p)^{\frac{1}{p}}$

Hölder’s inequality: $\Sigma_{k=1}^n |a_k b_k| \leq (\Sigma_{k=1}^n |a_k|^p)^{\frac{1}{p}} (\Sigma_{k=1}^n |b_k|^q)^{\frac{1}{q}}$ where $$p > 1$$, $$q > 1$$ and $$\frac{1}{q} + \frac{1}{q} = 1$$

Let $$f$$ be a mapping of one metric space $$X$$ into another metric space $$Y$$. $$f$$ continuous at the point $$x_0 \in X$$ if, given any $$\epsilon > 0$$, there exists a $$\delta > 0$$ such that $$\rho\prime(f(x), f(x_0) < \epsilon$$ wherever $$\rho(x, x_0) < \delta$$.

If $$f$$ bijective and both $$f$$ and $$f^{-1}$$ are continuous, $$f$$ is called a homeomorphism. $$X$$ and $$Y$$ are homeomorphic if there exists a homeomorphism between them.

Definition 2 A one-to-one mapping $$f$$ of one metric space $$R=(X, \rho)$$ onto another metric space $$R\prime = (Y, \rho\prime)$$ is said to be an isometric mapping (or isometry) if $\rho(x_1, x_2) = \rho\prime(f(x_1), f(x_2))$ for all $$x_1$$, $$x_2 in R$$. Correspondingly, $$R$$ and $$R\prime$$ are said to be isometric.

## Problems

1. Given a metric space $$(X, \rho)$$, prove that:

1. $$|\rho(x, z) - \rho(y, u)| \leq \rho(x, y) + \rho(z, u)$$ where $$x, y, z, u \in X$$

2. $$|\rho(x, z) - \rho(y, z)| \leq \rho(x, y)$$ where $$x, y, z \in X$$

Use the triangle inequality. Also take absolutes.

2. Verify that $(\Sigma_{k=1}^n a_k b_k)^2 = \Sigma_{k-1}^n {a_k}^2 \Sigma_{k-1}^n {b_k}^2 - \frac{1}{2}\Sigma_{i=1}^n\Sigma_{j=1}^n (a_i b_j - b_i a_j)^2$ and deduce the Cauchy-Schwarz inequality from this identity.

Trivial to deduce Cauchy-Schwarz from this identity, so just need to verify. Expand and plod through.

3. Verify that $(\int_a^b x(t) y(t) dt)^2 = \int_a^b x^2(t) dt \int_a^b y^2(t) dt - \frac{1}{2} \int_a^b \int_a^b [x(s)y(t) - y(s)x(t)]^2 ds dt$ and deduce Schwarz’s inequality from this identity.

Again, easy to deduce, just need to verify. Maybe use Lagrange identity and Fubini-Tonelli theorem?

4. What goes wrong in Example 10, page 41, if $$p < 1$$? Hint. Show that Minkowski’s inequality fails for $$p < 1$$.

Find counterexample.

5. Prove that the metric $\rho_0 (x, ) = \max_{1 \leq k \leq n} |x_k - y_k|$ is the limiting case of the metric $\rho_p(x, y) = (\Sigma_{k=1}^n |x_k - y_k|^p)^\frac{1}{p}$ in the sense that $\rho_0(x, y) = \max_{1 \leq k \leq n} |x_k - y_k| = \lim_{\rho \rightarrow \infty} (\Sigma_{k=1}^n |x_k - y_k|^p)^\frac{1}{p}$

The proof by CoveredInChocolate is great. Write out the sum. Choose the max of the $$|x_k - y_k|$$ and take it out as a factor. Then every other term tends to 0 as $$p$$ approaches infinity.

6. Starting from Young’s inequality $ab \leq \frac{a^p}{p} + \frac{b^q}{q}$ which applies for arbitrary positive $$a$$ and $$b$$ and whenever $$p > 1$$ and $$q > 1$$ and $\frac{1}{p} + \frac{1}{q} = 1$ deduce Hölder’s integral inequality.

Define $a = \frac{|x(t)|}{(\int_a^b |x(t)|^p dt)^{1/p}}, \quad b = \frac{|y(t)|}{(\int_a^b |y(t)|^q dt)^{1/q}}$ then apply Young’s inequality, and take integrals, noting that the right side is equal to $\frac{1}{p} + \frac{1}{q} = 1$ and then we’re almost there.

7. Use Hölder’s integral inequality to prove Minkowski’s integral inequality. $(\int_a^b|x(t) + y(t)|^p dt)^{1/p} \leq (\int_a^b|x(t)|^p)^{1/p} + (\int_a^b|y(t)|^p)^{1/p},\quad (p \geq 1)$

See CoveredInChocolate’s solution.

8. Exhibit an isometry between $$C_{[0,1]}$$ and $$C_{[1,2]}$$.

Too easy.