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Functional Analysis Exercises 1 : Distance Metrics

Avishek Sen Gupta on 21 September 2021

This post lists solutions to many of the exercises in the Distance Metrics section 1.1 of Erwin Kreyszig’s Introductory Functional Analysis with Applications. This is definitely a work in progress, and proofs may be refined or added over time.

1.1.2. Does \(d(x,y)={(x-y)}^2\) define a metric on the set of all real numbers?

Proof:

For the distance metric \(d(x,y)={(x-y)}^2\), we need to prove or disprove the Triangle Inequality:

\[d(x,z) \leq d(x,y) + d(y,z)\]

We start with the term \({(x-z)}^2\), as follows:

\[{(x-z)}^2 = {((x-y)+(y-z))}^2 \\ \Rightarrow {(x-z)}^2 = {(x-y)}^2 + {(y-z)}^2 + 2 (x-y) (y-z) \\ \Rightarrow d(x,z) = d(x,y) + d(y,z) + \underbrace{2 (x-y) (y-z)}_\text{positive or negative}\]

When the term \(2 (x-y) (y-z)\) is positive:

\[d(x,z) > d(x,y) + d(y,z)\]

When the term \(2 (x-y) (y-z)\) is negative or zero:

\[d(x,z) \leq d(x,y) + d(y,z)\]

Thus, the Triangle Inequality can only be satisfied for specific values of \(x\), \(y\), and \(z\); hence \(d(x,y)={(x-y)}^2\) is not a valid distance metric.

\[\blacksquare\]

1.1.3. Show that \(d(x,y)=\sqrt{|x-y|}\) defines a metric on the set of all real numbers.

Proof:

For the distance metric \(d(x,y)=\sqrt{\vert x-y \vert}\), we need to prove the Triangle Inequality:

\[d(x,z) \leq d(x,y) + d(y,z)\]

We start with the basic Triangle Inequality for \(\mathbb{R}\):

\[|x-z| \leq |x-y| + |y-z|\]

Adding and subtracting \(2 \sqrt{\vert x-y \vert \vert y-z \vert}\) on the RHS we get:

\[|x-z| \leq |x-y| + |y-z| + 2 \sqrt{\vert x-y \vert \vert y-x \vert} - 2 \sqrt{\vert x-y \vert \vert y-z \vert} \\ \Rightarrow |x-z| \leq {(\sqrt{\vert x-y \vert} + \sqrt{\vert y-z \vert})}^2 - \underbrace{2 \sqrt{\vert x-y \vert \vert y-z \vert}}_\text{positive}\]

Setting \(C=2 \sqrt{\vert x-y \vert \vert y-z \vert}>0\), we get:

\[|x-z| \leq {\left(\sqrt{\vert x-y \vert} + \sqrt{\vert y-z \vert}\right)}^2 - C \\ \Rightarrow {\left(\sqrt{\vert x-z \vert}\right)}^2 \leq {\left(\sqrt{\vert x-y \vert} + \sqrt{\vert y-z \vert}\right)}^2 - C \\ \Rightarrow {\left(\sqrt{\vert x-z \vert}\right)}^2 + C \leq {\left(\sqrt{\vert x-y \vert} + \sqrt{\vert y-z \vert}\right)}^2 \\ \Rightarrow {\left(\sqrt{\vert x-z \vert}\right)}^2 \leq {\left(\sqrt{\vert x-y \vert} + \sqrt{\vert y-z \vert}\right)}^2 \\ \Rightarrow \sqrt{\vert x-z \vert} \leq \sqrt{\vert x-y \vert} + \sqrt{\vert y-z \vert} \\ \Rightarrow d(x,z) \leq d(x,y) + d(y,z)\]

Hence, this proves the Triangle Inequality, and consequently, \(d(x,y)=\sqrt{\vert x-y \vert}\) is a valid distance metric.

\[\blacksquare\]

1.1.5. Let \(d\) be a metric on \(X\). Determine all constants \(k\) such that (i) \(kd\), (ii) \(d+k\) is a metric on \(X\).

(i) Proof:

Let \(\bar{d}(x,y)=kd(x,y)\) be a candidate metric on \(X\). For it to be a valid distance metric, it must satisfy the four axioms of a metric, i.e.:

Putting all of these together, we get the condition that \(k>0, k \in \mathbb{R}\).

(ii) Proof:

Let \(\bar{d}(x,y)=d(x,y) + k\) be a candidate metric on \(X\). For it to be a valid distance metric, it must satisfy the four axioms of a metric, i.e.:

Putting all of these together, we get the condition that \(k=0, k \in \mathbb{R}\).


1.1.6. Show that \(d\) in 1.1-6 satisfies the triangle inequality.

[TODO]


1.1.8. Show that another metric \(\bar{d}\) on the set \(X\) in 1.1-7 is defined by \(\bar{d}(x,y)=\int\limits_a^b |x(t) - y(t)| dt\).

[TODO]


1.1.9. Show that \(d\) in 1.1-8 is a metric.

[TODO]


1.1.10. (Hamming Distance) Let \(X\) be the set of all ordered triples of zeros and ones. Show that \(X\) consists of eight elements and a metric \(d\) on \(X\) is defined by \(d(x,y)=\) number of places where \(x\) and \(y\) have different entries. (This space and similar spaces of \(n\)-tuples play a role in switching and automata theory and coding. \(d(x,y)\) is called the Hamming distance between \(x\) and \(y\); cf. the paper by R. W. Hamming (1950) listed in Appendix 3.)

Proof:

The set of all ordered triples of zeros and ones, forms a sequence of numbers \(X=[0,7], x_i \in X, x_i \in \mathbb{Z}\) in their binary form. Thus, it is evident that the number of triples is \(2^3=8\).

Let \(a,b,c \in {0,1}\), then each number in this set, can be represented as \(x_i=a+2b+4c\). Then the suggested distance metric is:

\[d(x,y)=|a_x-a_y|+|b_x-b_y|+|c_x-c_y|\]

Let \(x_1\), \(x_2\), \(x_3\) be defined as follows:

\[x=a_x+2b_x+4c_x \\ y=a_y+2b_y+4c_y \\ z=a_z+2b_z+4c_z\] \[\begin{align*} d(x,z) &= |a_x-a_z|+|b_x-b_z|+|c_x-c_z| \\ &=|a_x-a_y+a_y-a_z|+|b_x-b_y+b_y-b_z|+|c_x-c_y+c_y-c_z| \end{align*}\]

Now we have the following inequalities:

\[|a_x-a_y+a_y-a_z| \leq |a_x-a_y|+|a_y-a_z| \\ |b_x-b_y+b_y-b_z| \leq |b_x-b_y|+|b_y-b_z| \\ |c_x-c_y+c_y-c_z| \leq |c_x-c_y|+|c_y-c_z|\]

Summing up these inequalities, and noting that the LHS resolves to \(d(x,z)\), we get:

\[d(x,z) \leq |a_x-a_y|+|a_y-a_z| + |b_x-b_y|+|b_y-b_z| + |c_x-c_y|+|c_y-c_z| \\ \Rightarrow d(x,z) \leq (|a_x-a_y|+|b_x-b_y|+|c_x-c_y|) + (|a_y-a_z|+|b_y-b_z|+|c_y-c_z|) \\ \Rightarrow d(x,z) \leq d(x,y) + d(y,z)\]

Thus, this proves the Triangle Inequality for the Hamming Distance as a metric.

\[\blacksquare\]

1.1.12. (Triangle inequality) The triangle inequality has several useful consequences. For instance, using the generalised triangle inequality, show that \(|d(x,y) - d(z,w)| \leq d(x,z) + d(y,w)\).

We write the two following triangle inequalities. One involves \(x\), \(y\), \(z\). The other one involves \(w\), \(y\), \(z\).

\[d(x,y) \leq d(x,z) + d(z,y)\] \[\begin{equation} d(z,y) \leq d(z,w) + d(w,y) \label{eq:1-1-12-1} \end{equation}\]

Adding \(d(x,z)\) to \(\eqref{eq:1-1-12-1}\), we get:

\[d(x,z) + d(z,y) \leq d(x,z) + d(z,w) + d(w,y)\]

Thus, we have:

\[d(x,y) \leq d(x,z) + d(z,y) \leq d(x,z) + d(z,w) + d(w,y) \\ \Rightarrow d(x,y) \leq d(x,z) + d(z,w) + d(w,y) \\ \Rightarrow d(x,y) \leq d(x,z) + d(z,w) + d(y,w) \text{(by the Symmetry property of a Distance Metric)}\] \[\begin{equation} d(x,y) - d(z,w) \leq d(x,z) + d(w,y) \label{eq:1-1-12-abs-1} \end{equation}\]

We write the two following triangle inequalities. One involves \(x\), \(z\), \(w\). The other one involves \(x\), \(y\), \(w\).

\[d(z,w) \leq d(z,x) + d(x,w)\] \[\begin{equation} d(x,w) \leq d(x,y) + d(y,w) \label{eq:1-1-12-2} \end{equation}\]

Adding \(d(z,x)\) to \(\eqref{eq:1-1-12-2}\), we get:

\[d(z,x) + d(x,w) \leq d(z,x) + d(x,y) + d(y,w)\]

Thus, we have:

\[d(z,w) \leq d(z,x) + d(x,w) \leq d(z,x) + d(x,y) + d(y,w) \\ \Rightarrow d(z,w) \leq d(z,x) + d(x,y) + d(y,w) \\ \Rightarrow d(z,w) \leq d(x,z) + d(x,y) + d(y,w) \text{(by the Symmetry property of a Distance Metric)}\] \[\begin{equation} d(z,w) - d(x,y) \leq d(z,x) + d(y,w) \label{eq:1-1-12-abs-2} \end{equation}\]

Summarising \(\eqref{eq:1-1-12-abs-1}\) and \(\eqref{eq:1-1-12-abs-2}\), we get:

\[d(x,y) - d(z,w) \leq d(x,z) + d(y,w) \\ d(z,w) - d(x,y) \leq d(x,z) + d(y,w) \\ \Rightarrow \mathbf{ |d(x,y) - d(z,w)| \leq d(x,z) + d(y,w) }\] \[\blacksquare\]

1.1.13. Using the triangle inequality, show that \(|d(x,z) - d(y,z)| \leq d(x,y)\).

Proof:

We have to show that:

\[|d(x,z) - d(y,z)| \leq d(x,y)\]

We write the following Triangle Inequality:

\[\begin{equation} d(x,z) \leq d(x,y) + d(y,z) \\ \Rightarrow d(x,z) - d(y,z) \leq d(x,y) \label{eq:1-1-13-abs-1} \end{equation}\]

The other Triangle Inequality we write is:

\[\begin{equation} d(y,z) \leq d(y,x) + d(x,z) \\ \Rightarrow d(y,z) - d(x,z) \leq d(y,x) \\ \Rightarrow d(y,z) - d(x,z) \leq d(x,y) \text{(by the Symmetry property of a Distance Metric)} \label{eq:1-1-13-abs-2} \end{equation}\]

Summarising the results of \(\eqref{eq:1-1-13-abs-1}\) and \(\eqref{eq:1-1-13-abs-2}\), we get:

\[d(x,z) - d(y,z) \leq d(x,y) \\ d(y,z) - d(x,z) \leq d(x,y) \\ \Rightarrow \mathbf{|d(x,z) - d(y,z)| \leq d(x,y)}\] \[\blacksquare\]

1.1.14. (Axioms of a metric) (M1) to (M4) could be replaced by other axioms (without changine the definition). For instance, show that (M3) and (M4) could be obtained from (M2) and \(d(x,y) \leq d(z,x) + d(z,y)\).

For reference, the axioms (M1) to (M4) are as follows:

Proof for (M3):

The allowed assumptions are:

Set \(z=y\) in (A2), so that we get:

\[\require{cancel} \begin{equation} d(x,y) \leq d(y,x) + \cancel{d(y,y)} \text{ (by (A1))} \\ \Rightarrow d(x,y) \leq d(y,x) \label{eq:1-1-14-abs-1} \end{equation}\]

From (A2), we get \(d(y,x)\) as:

\[\begin{equation} d(y,x) \leq d(z,y) + d(z,x) \label{eq:1-1-14-y-x} \end{equation}\]

Set \(z=x\) in \(\eqref{eq:1-1-14-y-x}\) again, so that we get:

\[\begin{equation} d(y,x) \leq d(x,y) + \cancel{d(x,x)} \text{ (by (A1))} \\ \Rightarrow d(y,x) \leq d(x,y) \\ \Rightarrow d(x,y) \geq d(y,x) \label{eq:1-1-14-abs-2} \end{equation}\]

Summarising the results of \(\eqref{eq:1-1-14-abs-1}\) and \(\eqref{eq:1-1-14-abs-2}\), we get:

\[d(x,y) \leq d(y,x) \\ d(x,y) \geq d(y,x)\]

This implies that:

\[\begin{equation} d(x,y)=d(y,x) \label{eq:1-1-14-symmetry} \end{equation}\] \[\blacksquare\]

Proof for (M4):

(M4) should immediately follow from \(\eqref{eq:1-1-14-symmetry}\), since:

\[d(x,y) \leq d(z,x) + d(z,y) \Rightarrow d(x,y) \leq d(x,z) + d(z,y)\] \[\blacksquare\]

1.1.15. Show that nonnegativity of a metric follows from (M2)to (M4).

Proof:

We have to prove that: \(d(x,y) \geq 0\) follows from (M2) and (M4).

From the Triangle Inequality (M4), we have:

\[d(x,y) \leq d(x,z) + d(z,y)\]

Set \(x=y\), then we get:

\[d(y,y) \leq d(y,z) + d(z,y) \\ \Rightarrow \underbrace{\cancel{d(y,y)}}_\text{by (M2)} \leq d(y,z) + \underbrace{d(y,z)}_\text{by (M3)} \\ \Rightarrow 2d(y,z) \geq 0 \\ \Rightarrow d(y,z) \geq 0\]

This proves (M1).

\[\blacksquare\]
tags: Mathematics - Proof - Functional Analysis - Pure Mathematics - Kreyszig