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algo_block_0 | \documentclass[a4paper]{book}
\usepackage{style}
\pagestyle{fancy}
\fancyhf{}\fancyfoot[LE,RO]{\thepage}
\fancyhead[RE]{\textit{\leftmark}}
\fancyhead[LO]{\textit{\rightmark}}
\fancyhead[LE]{Algorithms}
\fancyhead[RO]{Notes by Joachim Favre}
\fancypagestyle{plain}{\fancyhf{}\fancyfoot[LE,RO]{\thepage}}
\title{Alg... | algo | latex |
intro_to_ML_block_0 | \documentclass[a4paper]{article}
\usepackage{style}
\title{Introduction to machine learning --- BA3\\ Detailed summary}
\author{Joachim Favre\\ Course by Prof. Mathieu Salzmann }
\date{Autumn semester 2022}
\begin{document}
\maketitle
\cftsetindents{paragraph}{1.5em}{1em}
\setcounter{tocdepth}{5}
\tableofcontents... | intro_to_ML | latex |
intro_to_ML_block_1 | \begin{parag}{Supervised and unsupervised learning}
In \important{supervised learning}, we are given data and its groundtruth labels. The goal is, given new data, we want to predict new labels, by doing regression or classification.
In \important{unsupervised learning}, we are only given data (without any label), an... | intro_to_ML | latex |
intro_to_ML_block_2 | \begin{parag}{Regression and classification}
The goal of \important{regression} is to predict a continuous value for a given sample. The goal of \important{classification} is to output a discrete label (typically encoded in one-hot encoding with 0s and 1s or -1s and 1s).
The main difference is that there is the noti... | intro_to_ML | latex |
intro_to_ML_block_3 | \begin{parag}{Dimensionality reduction}
Dimensionality reduction has two main advantages.
The first one is that it allows to decrease the dimension of our data, which typically yield a tremendous speed-up while preserving a lot of the precision.
The second one is that, depending on the model, we can also map data... | intro_to_ML | latex |
intro_to_ML_block_4 | \begin{parag}{Notations}
We consider the following notation throughout this course, with some slight exceptions when specified otherwise. $N$ is the number of samples we have, $D$ is the dimensionality (the number of components) of any input, and $C$ is the dimensionality of any output.
Without specified otherwise,... | intro_to_ML | latex |
intro_to_ML_block_5 | \begin{parag}{Feature expansion}
Increasing the amount of dimensions from $D$ to $F$ of our input data may help our models (using non-linear functions, since they would be of no help). Thus, we may let the following function:
\[\phi\left(\bvec{x}\right) = \begin{pmatrix} 1 & x^{\left(1\right)} & \cdots & x^{\left(D\... | intro_to_ML | latex |
intro_to_ML_block_6 | \begin{subparag}{Remark}
The 1 we added to the input data to account for the bias is some kind of feature expansion.
\end{subparag} | intro_to_ML | latex |
intro_to_ML_block_7 | \begin{subparag}{Cover's Theorem}
Cover's theorem states (more or less) that doing non-linear feature expansion, then it is more likely for our data to be linearly separable.
\end{subparag}
\end{parag} | intro_to_ML | latex |
intro_to_ML_block_8 | \begin{parag}{Kernel}
We can notice that defining our $\phi$ functions for feature expansion can be really tedious. However, since most of our methods depend on a dot product of $\phi\left(\bvec{x}_i\right)^T \phi\left(\bvec{x}_j\right)$, which gives some kind of measure of similarity between $\bvec{x}_i$ and $\bvec{x... | intro_to_ML | latex |
intro_to_ML_block_9 | \begin{subparag}{Remark}
The main advantage of a kernel is that we don't need to know what function $\phi$ is linked to it.
\end{subparag} | intro_to_ML | latex |
intro_to_ML_block_10 | \begin{subparag}{Examples}
We can for instance use the \important{polynomial kernel}:
\[k\left(\bvec{x}_i, \bvec{x}_j\right) = \left(\bvec{x}_i^T \bvec{x}_j + c\right)^d\]
$c$ is often set to $1$ and $d$ to 2. For this kernel, the corresponding mapping $\phi$ is known. This is, except for multiplicative constants... | intro_to_ML | latex |
intro_to_ML_block_11 | \begin{parag}{Representer theorem}
The minimizer of a regularized empirical risk function can be represented as a linear combination of expanded features. In other words, we can write:
\[\bvec{w}^* = \sum_{i=1}^{N} \alpha_i^* \phi\left(\bvec{x}_i\right) = \Phi^T \bvec{\alpha}^*\]
where $\bvec{\alpha} \in \mathbb{R}... | intro_to_ML | latex |
intro_to_ML_block_12 | \begin{subparag}{Remark}
This theorem is really important to kernalise algorithms. When using it, the goal is to get rid of the $\Phi$ since we do not know the mapping $\phi$. Switching our view onto variables $\bvec{\alpha}$ instead of variables $\bvec{w}$ is typically a way to do so.
\end{subparag}
\end{parag} | intro_to_ML | latex |
intro_to_ML_block_13 | \begin{parag}{Loss function}
The \important{loss function} $\ell \left(\hat{\bvec{y}}_i, \bvec{y}_i\right)$ computes an error value between the prediction and the true value.
This is a measure of the error for any given prediction.
\end{parag} | intro_to_ML | latex |
intro_to_ML_block_14 | \begin{parag}{Empirical risk}
Given $N$ training samples $\left\{\left(\hat{\bvec{x}}_i, \hat{\bvec{y}}_i\right)\right\}$, the \important{empirical risk} is defined as:
\[R\left(\left\{\hat{\bvec{x}}_i\right\}, \left\{\hat{\bvec{y}}_i\right\}, W\right) = \frac{1}{N} \sum_{i=1}^{N} \ell \left(\hat{\bvec{y}}_i, \bvec{... | intro_to_ML | latex |
intro_to_ML_block_15 | \begin{subparag}{Regularised}
Sometimes, we want to regularise our objective function, so that we prevent weights to become to large and make a lot of overfitting. We then instead seek to minimise.
\[E\left(W\right) = R\left(W\right) + \lambda E_W\left(W\right)\]
where $\lambda$ is an hyperparameter and $E_W\left(W... | intro_to_ML | latex |
intro_to_ML_block_16 | \begin{parag}{Gradient descent}
The goal of \important{gradient descent} is to minimise a function (an empirical risk $R\left(W\right)$ in this context). The idea is to begin with an estimate $W_0$ (typically completely random), and then to update it iteratively, by following the direction of greatest decrease (the op... | intro_to_ML | latex |
intro_to_ML_block_17 | \begin{subparag}{Remark}
This algorithm does not always converge and, when it does, not necessarily to a minimum nor to the global minimum.
\end{subparag} | intro_to_ML | latex |
intro_to_ML_block_18 | \begin{parag}{Evaluation metrics}
Once a supervised machine learning model is trained, we want to be able to understand how well it performs on unseen test data (which must absolutely be separated from the train data).
We could use the loss function, but we may also use a different one.
For regression, we typical... | intro_to_ML | latex |
intro_to_ML_block_19 | \begin{subparag}{Remark}
There are many more metrics for regression and classification. For the former, we could quote RMSE (root mean squared error), MAE (mean absolute error) or the MAPE (mean absolute percentage error). For the latter, making a confusion matrix or computing the AUC (area under the ROC curve) can gi... | intro_to_ML | latex |
intro_to_ML_block_20 | \begin{parag}{Decision boundary}
A classifier leads to a decision boundary. This is an object of dimension $D-1$ (a line if our data lies on a plane for instance), which splits the space into two regions: one where samples are considered positive (the predicted value is closer to the value of positive samples), and on... | intro_to_ML | latex |
intro_to_ML_block_21 | \begin{subparag}{Remark}
A classifier is said to be linear if its decision boundary is an hyperplane (a straight line if the data lies on a plane for instance).
\end{subparag}
\end{parag} | intro_to_ML | latex |
intro_to_ML_block_22 | \begin{parag}{Margin}
If $C = 1$, the orthogonal distance between the decision boundary and the nearest sample is called the margin.
\imagehere[0.5]{margin.png}
\end{parag} | intro_to_ML | latex |
intro_to_ML_block_23 | \begin{parag}{Overfitting}
When we increase the complexity of the model (by changing the hyperparameters) we get better and better result for both training and test data. However, there is a point at which increasing the complexity keeps decreasing error on training data but increases the error on test data. This is a... | intro_to_ML | latex |
intro_to_ML_block_24 | \begin{parag}{Cross-validation}
Cross-validation is a way to find good hyperparameters that prevent overfitting. We test different models (in a predefined set), assign to each of them an error value, and pick the one yielding the smallest error.
The idea of \important{$k$-fold cross-validation} is, to evaluate the e... | intro_to_ML | latex |
intro_to_ML_block_25 | \begin{subparag}{Remark}
Note that leaving $k = N$ (where $N$ is the number of training samples) is also sometimes named \important{leave-one-out cross-validation}. This is really expensive but wastes (almost) no data.
Another way to do cross-validation, which is much cheaper, is to split our training data into trai... | intro_to_ML | latex |
intro_to_ML_block_26 | \begin{parag}{Data representation}
All the models we will see only work for fixed size data. If we want to handle data of varying size (such as text or pictures), a good way is to consider \important{bag of words}: consider the number of times each word from a dictionary appears in the text and put this as a big vecto... | intro_to_ML | latex |
intro_to_ML_block_27 | \begin{parag}{Pre-processing}
The training data might have problems: it might have noise (because of measurement errors), incorrect values, and so on. To fix those, a good idea is to do pre-processing. | intro_to_ML | latex |
intro_to_ML_block_28 | \begin{subparag}{Normalisation}
To begin with, a good idea is to scale each individual feature dimension to fall within a specified range (so that we don't give more impact to a dimension ranging from 1000 to 10000 than to another dimension ranging from 0 to 1). This can typically be done by using \important{normalisa... | intro_to_ML | latex |
intro_to_ML_block_29 | \begin{subparag}{Imbalanced data}
Another important thing to consider is \important{imbalanced data}. There might be 10 times as much data in one class as in another (between-class imbalance), or data inside a class might have a lot of representative at some point in space and much less at other points (within-class i... | intro_to_ML | latex |
intro_to_ML_block_30 | \begin{parag}{Ridge regression}
The output of \important{ridge regression} is computed by a simple dot product:
\[\hat{\bvec{y}}_i = W^T \phi\left(\bvec{x}_i\right)\]
The training objective function we want to minimise is a squared Euclidean distance regularised by the sum of squares of the weights:
\[E\left(W\ri... | intro_to_ML | latex |
intro_to_ML_block_31 | \begin{subparag}{Linear regression}
Leaving $\lambda = 0$, we get the special case of \important{linear regression}. Then, the closed-form formula can be rephrased as:
\[W^* = \left(\Phi^T \Phi\right)^{-1} \Phi^T \bvec{y} = \Phi^{\dagger} Y\]
where $\Phi^{\dagger}$ is the Moore-Penrose pseudo-inverse.
\end{subpara... | intro_to_ML | latex |
intro_to_ML_block_32 | \begin{subparag}{Kernelisation}
Using the representer theorem, we can find that:
\[A^* = \left(K + \lambda I_N\right)^{-1} Y\]
This is not of much use on its own, but we can use this result to find how we predict a value $\hat{\bvec{y}}$ for a new $\bvec{x}$:
\[\hat{\bvec{y}} = Y^T \left(K + \lambda I_N\right)^... | intro_to_ML | latex |
intro_to_ML_block_33 | \begin{subparag}{Classification}
Ridge regression can be used for classification tasks, by inputting the result into a softmax function (defined right after), but this does not work very well because we are not encoding this in the objective function.
This is named a \important{least-square classifier}.
\end{subpar... | intro_to_ML | latex |
intro_to_ML_block_34 | \begin{parag}{Logistic regression}
In \important{logistic regression} (which is a linear classification algorithm), we consider negative samples to be $y_i = 0$.
The output of logistic regression is computed by using the \important{softmax function}:
\[\hat{y}^{\left(k\right)} = \frac{\exp\left(\bvec{w}^T_{\left(k... | intro_to_ML | latex |
intro_to_ML_block_35 | \begin{subparag}{One dimension}
Let's consider $C = 1$. This special case of the softmax function is named the \important{logistic function}:
\[\hat{y}\left(\bvec{x}\right) = \sigma\left(\bvec{w}^T \bvec{x}\right) = \frac{1}{1 + \exp\left(- \bvec{w}^T \bvec{x}\right)}\]
One-dimensional cross-entropy can be rewritte... | intro_to_ML | latex |
intro_to_ML_block_36 | \begin{subparag}{Kernelisation}
This algorithm can be kernalised even though this is not very common.
\end{subparag}
\end{parag} | intro_to_ML | latex |
intro_to_ML_block_37 | \begin{parag}{Support vector machine}
In \important{support vector machine} (SVM) classification (which is also a linear classifier), we consider negative samples to be $y_i = -1$. Also, we leave $\widetilde{\bvec{w}}$ to be the vector of parameters without $w^{\left(0\right)}$, and $\bvec{x} \in \mathbb{R}^D$ to not ... | intro_to_ML | latex |
intro_to_ML_block_38 | \begin{subparag}{Support vectors}
We notice that, for the margin to be maximised, there must be at least a point from each class lying on it. Such points are named \important{support vectors}.
\end{subparag} | intro_to_ML | latex |
intro_to_ML_block_39 | \begin{subparag}{Hinge loss}
By rewriting the constraints, we get:
\[\xi_i \geq 1 - y_i\left(w^{\left(0\right)} + \widetilde{\bvec{w}}^T \bvec{x}_i\right)\]
For samples $i$ that satisfy the support vector machine problem (they are on not in the margin nor misclassified), we have $\xi_i = 0$ (since they are forced... | intro_to_ML | latex |
intro_to_ML_block_40 | \begin{subparag}{Dual problem}
We can reformulate our problem by letting one variable per training sample (meaning that we have $N$ variables instead of $\left(D+1\right)$):
\[\argmax_{\left\{\alpha_i\right\}} \left(\sum_{i=1}^{N} \alpha_i - \frac{1}{2} \sum_{i=1}^{N} \sum_{j=1}^{N} \alpha_i \alpha_j y_i y_j \bvec{x... | intro_to_ML | latex |
intro_to_ML_block_41 | \begin{subparag}{Kernelisation}
We need the dual problem to kernelise the SVM:
\[\argmax_{\left\{\alpha_i\right\}} \left(\sum_{i=1}^{N} \alpha_i - \frac{1}{2} \sum_{i=1}^{N} \sum_{j=1}^{N} \alpha_i \alpha_j y_i y_j k\left(\bvec{x}_i, \bvec{x}_j\right)\right),\]
subject to $\sum_{i=1}^{N} \alpha_i y_i = 0$ and $0 \l... | intro_to_ML | latex |
intro_to_ML_block_42 | \begin{subparag}{Multi-class SVM}
To generalise our algorithm to multiple class, we can use multiple ways. The idea is always to use several binary classifiers.
One way is to use \important{one-vs-rest}: we train classifiers stating if the component is in class $i$ or not. Another way is to use \important{one-vs-one... | intro_to_ML | latex |
intro_to_ML_block_43 | \begin{subparag}{Primal derivation}
Let's consider how we got the formula for the primal model, since it may typically help to remember and understand it.
First, we know that any two points on the decision boundary have the same prediction (which is 0), which yields that:
\[0 = \hat{y}_1 - \hat{y}_2 = \left(w^{\le... | intro_to_ML | latex |
intro_to_ML_block_44 | \begin{parag}{$K$-nearest neighbours}
The idea of \important{$k$-nearest neighbours} (kNN) is to compute the distance between the test sample $\bvec{x}$ and all training samples $\left\{\bvec{x}_i\right\}$ and find the $k$ samples with minimum distances. Then, we can do classification by finding the most common label ... | intro_to_ML | latex |
intro_to_ML_block_45 | \begin{subparag}{Remark}
The result of this model depends on the choice of the distance function. One can take the \important{Euclidean distance}:
\[d\left(\bvec{x}_i, \bvec{x}\right) = \sqrt{\sum_{d=1}^{D} \left(x_i^{\left(d\right)} - x^{\left(d\right)}\right)^2}\]
However, for some other structures such as hist... | intro_to_ML | latex |
intro_to_ML_block_46 | \begin{subparag}{Curse of dimensionality}
Because of a principle named the \important{curse of dimensionality}, we need exponentially more points to cover a space as the number of dimensions increases. Using dimensionality reduction is a good idea with this algorithm.
\end{subparag} | intro_to_ML | latex |
intro_to_ML_block_47 | \begin{subparag}{Complexity}
Unlike most of the other models, increasing the hyperparameter of this model (the $k$) leads to decreased complexity: the higher the $k$, the less complex the decision boundary is and thus the less overfit we have.
\end{subparag}
\end{parag} | intro_to_ML | latex |
intro_to_ML_block_48 | \begin{parag}{Neural networks}
Neural networks can do both classification and regression (depending on the output representation and the empirical risk used, typically square loss for regression and cross-entropy for classification), and their main advantage is that they learn a good model during the training.
This ... | intro_to_ML | latex |
intro_to_ML_block_49 | \begin{subparag}{Activation functions}
There are multiple choice for activation functions. The important thing is that they are non-linear.
We can for instance take the ReLU (Rectified Linear Unit) activation function:
\begin{functionbypart}{f\left(a\right)}
a, & \text{if } a > 0 \\
0, & \text{otherwise}
\end{f... | intro_to_ML | latex |
intro_to_ML_block_50 | \begin{subparag}{Convolutional network}
When working with pictures, just vectorising them may give a lot of elements while removing the fact that the picture is inherently two-dimensional. A way to circumvent this problem is using convolutions.
To make a convolution, we need a small matrix of elements (plus a bias).... | intro_to_ML | latex |
intro_to_ML_block_51 | \begin{parag}{Principle component analysis}
Sometimes, we realise that data is given in many dimensions, but actually lies in many less dimensions. The idea of \important{principle component analysis} (PCA) is to project the data on some orthogonal axis (of lower dimension), in a way to maximise the kept variance.
L... | intro_to_ML | latex |
intro_to_ML_block_52 | \begin{subparag}{Remark}
Since the axis on which we project the data are orthogonal, we have:
\[W^T W = I_d\]
To make sure of this, we need to take the eigenvector so that they are orthogonal. This can always be done because $C$ is symmetric, thanks to the spectral theorem (this theorem also allows to know that we... | intro_to_ML | latex |
intro_to_ML_block_53 | \begin{subparag}{Mapping}
From our computation, we can notice that, for any point $\bvec{y} \in\mathbb{R}^d$, we can move it to the high-dimensional space:
\[\hat{\bvec{x}} = \bar{\bvec{x}} + W \bvec{y}\]
This yields all the advantages presented in the first section of this document.
\end{subparag} | intro_to_ML | latex |
intro_to_ML_block_54 | \begin{subparag}{Kerenelisation}
PCA can be kernalised in a non-trivial fashion.
First, we need to account for the fact that our data may not be centered in input-space (this was done above by considering the mean of our input values), letting:
\[\widetilde{K} = K - 1_N K - K 1_N + 1_N K 1_N\]
where $1_N$ is an $... | intro_to_ML | latex |
intro_to_ML_block_55 | \begin{parag}{Autoencoder}
Another way to do dimensionality reduction is through a neural network.
The idea is to have a double funnel shaped neural network: an encoder decreasing the dimension, a layer with $d$ neurons, and a decoder increasing back the number of dimensions.
\imagehere[0.7]{Autoencoder.png}
We c... | intro_to_ML | latex |
intro_to_ML_block_56 | \begin{subparag}{Remark}
This can also be used to both do dimensionality reduction and mapping data back from the low-dimensional space.
\end{subparag} | intro_to_ML | latex |
intro_to_ML_block_57 | \begin{subparag}{Convolutional autoencoder}
We can use convolutional neural networks for autoencoders. To do so, we use the inverse functions of those convolutions.
\end{subparag}
\end{parag} | intro_to_ML | latex |
intro_to_ML_block_58 | \begin{parag}{Fisher linear discriminant analysis}
Even though \important{Fisher linear discriminant analysis} (LDA) is a dimensionality reduction algorithm, it is a supervised learning one. Its goal is to project data on lower space, while keeping classes (hence the supervision) clustered. It considers that clusterin... | intro_to_ML | latex |
intro_to_ML_block_59 | \begin{parag}{$K$-means clustering}
The idea of \important{$k$-means clustering} is to consider that clustering should also be done relatively to compactness.
To do so, we consider $K$ (an hyperparameter) cluster centers $\left\{\bvec{\mu}_1, \ldots, \bvec{\mu}_K\right\}$. While we are not converged, we assign each ... | intro_to_ML | latex |
intro_to_ML_block_60 | \begin{subparag}{Hyperparameter}
To choose the $K$, a good way is to draw a graph of the average within-cluster sum of distances with respect to the number of cluster, and pick a point at its ``elbow'' (where the drop in the $y$-axis becomes less significant).
\end{subparag}
\end{parag} | intro_to_ML | latex |
intro_to_ML_block_61 | \begin{parag}{Spectral clustering}
The idea of \important{spectral clustering} is to consider that clustering should be done relatively to connectivity instead: we group the points based on edges in a graph, and remove some of the edges with longest length. Let us first consider the case where we only want to make 2 c... | intro_to_ML | latex |
intro_to_ML_block_62 | \begin{subparag}{Remark}
Since we had to relax the problem, the solution is not always optimal.
\end{subparag} | intro_to_ML | latex |
intro_to_ML_block_63 | \begin{subparag}{$K$-way partitions}
To obtain more than two clusters, we have two choices.
The first one is to recursively apply the two-way partitioning algorithm. This is inefficient and unstable.
The second one is to find $K$ eigenvectors. This leads to each point being represented by a $K$-dimensional vector.... | intro_to_ML | latex |
analysis_1_chunk_0 | MATH 101 (en)– Analysis I (English)
Notes for the course given in Fall 2021
Teacher:Roberto Svaldi
Head Assistant: Stefano Filipazzi
Notes written by Zsolt Patakfalvi & Roberto Svaldi
Thursday 12th October, 2023
This work is licensed under a Creative Commons “Attribution-
NonCommercial-NoDerivatives 4.0 International” ... | analysis_1 | pdf |
analysis_1_chunk_1 | CONTENTS
1
Proofs
3
2
Basic notions
8
2.1
Sets
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.2
Number sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.2.1
Half lines, intervals, balls
. . . . . . . . . . . . . . . . . . . . . . . . . .
10
2.2... | analysis_1 | pdf |
analysis_1_chunk_2 | 22
2.4.3
Rational numbers are dense in R . . . . . . . . . . . . . . . . . . . . . .
23
2.4.4
Irrational numbers are dense in R
. . . . . . . . . . . . . . . . . . . . .
25
2.5
Absolute value
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.5.1
Properties of the absolute value . . . . . . . ... | analysis_1 | pdf |
analysis_1_chunk_3 | 42
4.2
Induction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
4.3
Bernoulli inequality and (non-)boundedness of geometric sequences . . . . . . .
45
4.4
Limit of a sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
4.4.1
Limits and algebra . . . . . . . .... | analysis_1 | pdf |
analysis_1_chunk_4 | 1
PROOFS
The means to explore analysis from a mathematical viewpoint within this course will be
mathematical proofs. Part of the goal of the course will be for you to learn how to prove
mathematical statements via mathematical proofs.
There are two main types of proof that we will encounter:
◦Constructive proof: an arg... | analysis_1 | pdf |
analysis_1_chunk_5 | ◦p is prime;
◦if a, b are natural numbers and p divides ab, then either p divides a or p divides b.
Proof of Proposition 1.1. Assume that
√
3 is rational. Thus, we may write
√
3 = a
b
(1.2.a)
for some integers a and b ̸= 0. As
√
3 > 0, a and b should have the same sign. If they are
both negative, by multiplying both by... | analysis_1 | pdf |
analysis_1_chunk_6 | that a and b are relatively prime, that is, they do not share any prime factors. Multiplying
both sides of (1.2.a) by b, then, since b ̸= 0,
b
√
3 = a.
(1.2.b)
Squaring both sides of (1.2.b) yields
b2 · 3 = a2.
(1.2.c)
Hence, as 3 divides the left hand side of (1.2.c), 3 must divide the right hand side, too. Thus,
a = ... | analysis_1 | pdf |
analysis_1_chunk_7 | to be extremely careful: it is indeed very easy to write wrong proofs! This is often do to that
the fact that one may assume something wrong in the course of a proof: if the premise of an
implication is false, then anything can follow from it.
Example 1.5. Here is an example of an (incorrect) proof showing that 1 is th... | analysis_1 | pdf |
analysis_1_chunk_8 | This claim cannot possibly be true: in fact, 2 is definitely an integer and 2 > 1. Even better,
the set of integeral numbers is not bounded from above2, that is, there is no real number C such
that z ≤C for all z ∈Z.
What went wrong in the above proof? All the algebraic manipulations that we made following
the first line... | analysis_1 | pdf |
analysis_1_chunk_9 | we can represent numerically by writing down a decimal expansion, for example,
√
2 =1.414213562373095048801688724209698078569671875376948073176679737990
7324784621070388503875343276415727350138462309122970249248360 . . . .
As it suggested from this example, it may be the case that when we try to represent certain
real ... | analysis_1 | pdf |
analysis_1_chunk_10 | 3See Section 3 for the definition and basic properties of complex numbers.
4Some of the most important classes of functions that we will encounter are those of continuous, differentiable,
integrable, analytic functions, but there are many more other possible classes of functions that are heavily studied
in analysis
5The ... | analysis_1 | pdf |
analysis_1_chunk_11 | Proposition 1.6. 0.¯9 = 1
By 0.¯9 we denote the real number whose decimal representation is given by an infinite
sequence of 9 in the decimal part, 0.999999 . . ..
Proof. We give two proofs none of which is completely correct, at least as far as our current
definition and knowledge of the real numbers go. Nevertheless, w... | analysis_1 | pdf |
analysis_1_chunk_12 | way from here, nevertheless we continue the argument for completeness. If you are not
comfortable with it now, it is completely OK, just skip this part of the proof.
However, before we proceed, we need to show an identity for the sum of elements in a
geometric series6.
Claim. Let a ∈R, a ̸= 1. Then,
a + a2 + · · · + an... | analysis_1 | pdf |
analysis_1_chunk_13 | And then we can proceed showing the statement:
∞
X
i=1
9
10i = 9 ·
∞
X
i=1
1
10i = 9 · lim
n→∞
n
X
i=1
1
10i
!
=
9 · lim
n→∞
1
10 −
1
10n+1
1 −1
10
!
= 9 ·
1
10 −lim
n→∞
1
10n+1
1 −1
10
=
9 ·
1
10
1 −1
10
= 91
9 = 1.
In Section 2 and in the following one, we will introduce all the necessary tools, definitions,
notatio... | analysis_1 | pdf |
analysis_1_chunk_14 | 2
BASIC NOTIONS
2.1
Sets
A set S is a collection of objects called elements. If a is an element of S, we say that a
belongs to S or that S contains a, and we write a ∈S. If an element a is not in S, we then
write a ̸∈S. If the elements a, b, c, d, . . . form the set S, we write S = {a, b, c, d, . . . }. We can
also defi... | analysis_1 | pdf |
analysis_1_chunk_15 | write T ⊊S.
Example 2.4. 2N ⊊N since 1 ̸∈2N.
If we just write T ⊆S, we mean that T is a subset of S that may be equal to S, but we are
not making any particular statement about whether or not T is a strict subset of S. Hence, in
the previous Example 2.4, we may have also used the notation 2N ⊆N and that would have
been... | analysis_1 | pdf |
analysis_1_chunk_16 | 2.2
Number sets
There are a few important sets that we are going to work with all along this course:
(1) ∅: the empty set; it is the set which has no elements, ∅:= { }.
Exercise 2.6. Show that for any set S, ∅⊆S.
(2) N : the set of natural numbers, N := {0, 1, 2, 3, 4, 5, 6, . . . }.
N is well ordered, that is, all its... | analysis_1 | pdf |
analysis_1_chunk_17 | defined a multiplicative inverse x−1 such that x · x−1 = 1;
◦the world ordered refers to the fact that given two elements x, y ∈R we can always
decide whether x < y, or x > y, or x = y; moreover, this comparison is also
compatible with the operations that make R into a field.
(3) R satisfies the Infimum Axiom 2.22, that wi... | analysis_1 | pdf |
analysis_1_chunk_18 | 2.2.1
Half lines, intervals, balls
We introduce here further notation regarding the real numbers and some special classes of
subsets that we will be using all throughout the course.
(1) Invertible real numbers: R∗:= {x ∈R | x ̸= 0}.
(2) Closed half lines: R+ := {x ∈R | x ≥0}, R−:= {x ∈R | x ≤0}.
At times, these are als... | analysis_1 | pdf |
analysis_1_chunk_19 | a as
B(a, δ) :=]a −δ, a + δ[.
(6) Closed balls: let a, δ ∈R, δ ≥0; we define the closed ball B(a, δ) ⊆R of radius δ and
center a as
B(a, δ) := [a −δ, a + δ].
When δ = 0, then B(a, 0) = {a}.
2.2.2
Extended real numbers
The extended real line is the set
R := {−∞, +∞} ∪R.
The symbol +∞(resp. −∞) is called “plus infinity” (r... | analysis_1 | pdf |
analysis_1_chunk_20 | involving ±∞; thus, be very careful not to treat those as numbers. If you think carefully a bit,
you can see that it is hard to coherently define for example the result of the addition
+∞+ (−∞).
Later in the course we will use extensively these symbols. For the time being, we just want
to use them to define the following... | analysis_1 | pdf |
analysis_1_chunk_21 | for all s ∈S.
(2) If S has an upper (resp. a lower) bound then S is said to be bounded from above (resp.
bounded from below).
(3) The set S is said to be bounded if it is bounded both from above and below.
For a set S ⊆R in general upper and lower bounds are not unique.
Example 2.9.
(1) The set N ⊂R is bounded from bel... | analysis_1 | pdf |
analysis_1_chunk_22 | (3) The set S := {n2|n ∈Z} is bounded from below: in fact, ∀n ∈N, n2 ≥0, thus 0 is
a lower bound. On the other hand, it is not bounded. In fact, assume for the sake of
contradiction that S were bounded from above, i.e., that there exists u ∈R and u ≥s,
∀s ∈S. Since for any n ∈N, n2 ≥n, then it would follow that u > n, ... | analysis_1 | pdf |
analysis_1_chunk_23 | [3, 5], ]3, 5], ]3, 5[.)
Using the discussion of the above examples, we summarize here some of the main properties
of upper and lower bounds.
Proposition 2.10. Let S ⊂R be a non-empty set. Let c ∈R.
(1) If u is an upper bound for S, then for any d ≥u, d is also an upper bound for S.
(2) If l is a lower bound for S, the... | analysis_1 | pdf |
analysis_1_chunk_24 | (3) If c is a lower bound for S, then c ≤s for all element s ∈S. Since T ⊆S, this means
that any element t ∈T is also an element of S. Hence, a fortiori, the inequality c ≤s,
∀s ∈S implies also that c ≤t, ∀t ∈T.
The case of an upper bound is analogous, it suffices to change the verse of the inequalities.
(4) Since T is n... | analysis_1 | pdf |
analysis_1_chunk_25 | for S. To conclude we need to show that no real number c > a (resp. d < b) is a lower
bound (resp. an upper bound) of S. To show this, it suffices to show that there exists an
element m ∈S such that m < c. Since c > a, then a < a + c−a
2
< c. If a + c−a
2
∈S, it
suffices to take m := a + c−a
2 . If a + c−a
2
̸∈S, then a + ... | analysis_1 | pdf |
analysis_1_chunk_26 | n, n′ are distinct, i.e., n ̸= n′, we can assume that n < n′. As n′ is a maximum, then n′ ∈S.
But as n is also a maximum, in particular, n is also an upper bound, i.e., n ≥s, ∀s ∈S; hence,
also n ≥n′, which is in contradiction with our assumption above that n′ > n.
You can apply a similar argument for the uniqueness of... | analysis_1 | pdf |
analysis_1_chunk_27 | consequence of Definition 2.11 and of Proposition 2.10
Proposition 2.15. Let S ⊆R be a bounded interval of extremes a < b.
(1) The maximum of S exists if and only if b ∈S. In this case, max S = b.
(2) The minimum of S exists if and only if a ∈S. In this case, min S = b.
When S is not an interval, it may be more complica... | analysis_1 | pdf |
analysis_1_chunk_28 | n−1
n
| n ∈Z∗
+
, as it is the least of all possible upper bounds. On the other hand, 1 cannot
be the maximum of S as 1 ̸∈S. This phenomenon motivates the next definition.
Definition 2.17. Let S ⊆R be a non-empty subset.
14 | analysis_1 | pdf |
analysis_1_chunk_29 | (1) If the set U of all upper bounds of S is non-empty and U admits a minimum u ∈U, then
we call u the supremum of S.
(2) If the set L of all lower bounds of S is non-empty and L admits a maximum l ∈L, then
we call l the supremum of S.
Remark 2.18. Let S ⊆R be a non-empty subset.
If the set U of all upper bounds of S i... | analysis_1 | pdf |
analysis_1_chunk_30 | You can apply a similar argument for the uniqueness of the minimum.
Example 2.21.
(1) Let S :=
n−1
n
n ∈Z∗
+
. Then, sup S = 1, cf. Example 2.16.3.
(2) Take S := {n3|n ∈Z}. Then, S is unbounded. Thus, inf S, sup S do not exist.
(3) If S is a bounded interval of extremes a < b, then
sup S = b,
inf S = a.
Indeed, ... | analysis_1 | pdf |
analysis_1_chunk_31 | Axiom 2.22. [Infimum axiom] Each non-empty subset S of R∗
+ admits an infimum (which
is a real number).
Remark 2.23. In Mathematics, an axiom is a statement that we are going to assume to be true,
without requiring for it a formal proof. When we introduce an axiom, we are free to use the
properties stated in the axiom, ... | analysis_1 | pdf |
analysis_1_chunk_32 | we know that if such l existed, then l <
√
3, since
√
3 ̸∈Q, cf Proposition 2.38, and l is certainly
a lower bound for l. But then, Proposition 2.44 shows that there exists a rational number m
such that l < m <
√
3. As m <
√
3, then we know that m is also a lower bound for S. This is
clearly a contradiction, as m ∈Q na... | analysis_1 | pdf |
analysis_1_chunk_33 | Let W ⊆R be the subset obtained by translating the elements of S by −l + 1,
W := {s −l + 1 | s ∈S}.
Why did we choose to translate the elements of S by −l+1? The reason is that W ⊆R∗
+:
in fact, by (2.26.a), s−l+1 ≥1 > 0, for all s ∈S.9 As W ⊆R∗
+, the Infimum Axiom 2.22
implies that inf W exists, call it a := inf W. Th... | analysis_1 | pdf |
analysis_1_chunk_34 | S′ := {−x | x ∈S}.
Since S is bounded from above, then S′ is bounded from below. [Prove this!] Then by
part (1), inf S′ exists. It is left to you to show that sup S = −inf S′.
We have seen the definition of infimum/supremum and minimum/maximum.
Both the
infimum (resp. supremum) and minimum (resp. maxima) of a set S, provi... | analysis_1 | pdf |
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