# Analysis

### What is real analysis?

In this article, we answer the question: what is real analysis about? Apparently the word real comes from the set...

### The optimal step gradient method

In this article, we propose an exercise that describes the optimal method of the step gradient for coercive functions on...

### Translation operator

We discuss the properties of the translation operator defined in a Lebesgue space. The latter appears in the study of...

### Lyapunov stability for nonlinear systems

We discuss the Lyapunov stability for nonlinear systems. Indeed, having a reference solution, a stationary solution, one wonders if the...

### Instability of solutions to nonlinear systems

In this post, we propose two results in the instability of solutions to nonlinear systems. Here, we study ordinary differential...

### Open mapping theorem in functional analysis

In this article, we give an application of the open mapping theorem in functional analysis. This fundamental theorem in functional...

### Differential calculus in Banach spaces

Differential calculus in Banach spaces is a very important part of mathematics. In fact, the treatment of partial differential equations...

### Heat equation using the Fourier series

In this article, we learn how to solve the heat equation using the Fourier series. The heat equation belongs to...

### Gronwall lemma

In this article, we state and prove Gronwall lemma and give some of its applications. In fact, we will use...

### Stability analysis of differential equations

The stability analysis of solutions of differential equations is one of the most important axes in ODE. Here w gives...
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