Articles of analysis

For any subset $S$ of $\mathbb{R}$, the distance function $d_s(x) = d(x,S)$ is continuous

Let $S \subset \mathbb R$ be any set, and define for any $x \in \mathbb R$ the distance between $x$ and the set $S$ by $d(x,S) = \inf\{|x-s| : s \in S\}$. Prove that the function $d_s: \mathbb R \to [0,+\infty)$ given by $d_s(x) = d(x,S)$, is Lipschitz continuous. Prove that if $S$ is compact […]

How to solve such a nonlinear ODE, the analytical solution of which is known!

I have the following ODE with initial/boundary value conditions: $$\left. \begin{aligned} \left(x^2-10 x-y^2\right)y\, y'(x)+(x-5) y^2 y'(x)^2-(x-5) y^2=0 ;\qquad (\text{ODE})\\ y(0)^2=25;\qquad y'(0)^2=\frac{3-\sqrt{5}}{2} \qquad\qquad\qquad (\text{IBCs}) \end{aligned} \right\} $$ How to solve such a nonlinear ODE? Additionally, how can I verify whether a special function is a potential solution or not, e.g., the one in implicit form as […]

How prove this mathematical analysis by zorich from page 233

Let $f$ be twice differentiable on an interval $I$,Let $$M_{0}=\sup_{x\in I}{|f(x)|},M_{1}=\sup_{x\in I}{|f'(x)|},M_{2}=\sup_{x\in I}{|f”(x)|}$$ show that (a):$$M_{1}\le 2\sqrt{M_{0}M_{2}}$$ if the length of $I$ is not less than $2\sqrt{\dfrac{M_{0}}{M_{2}}}$ (b):the numbers $2$ and $\sqrt{2}$ (in part a) cannot be replaced by smaller numbers. My try:for part $(a)$ I can prove if the length of $I$ is not […]

Does projection onto a finite dimensional subspace commute with intersection of a decreasing sequence of subspaces: $\cap_i P_W(V_i)=P_W(\cap_i V_i)$?

Let $V$ to be an infinite dimensional linear space over some field $k$. (you can take $k=\mathbb{C}$, or further assume $V$ is a complex Hilbert space). And assume $W$ is a finite dimensional subspace of $V$, and denote the projection from $V$ to $W$ by $P_W$. Let $V_1\supseteq V_2\supseteq V_3\supseteq\cdots$ be a decreasing sequence of […]

How can I show this function is Weakly Sequentially Lower Semicontinuos?

Suppoe $X$ is a Banach spaces and $G\subset X$ is a convex open set. Let $\phi:G\rightarrow \mathbb{R}$ be a $C^1$ function and assume that $\phi’$ is a bounded and peseudo-monotone map (see here for a definiton of pseudo-monotone). We say that $\phi$ is weakly sequentially lower semicontinuos (WSLSC) in $G$ if for every sequence $x_n$ […]

Proofing that the exponential function is continuous in every $x_{0}$

Given: $$\exp: \mathbb{R} \ni x \mapsto \sum_{k=0}^{\infty } \frac{1}{k!} x^{k} \in \mathbb{R}$$ also $e = \exp(1)$. For all $x \in \mathbb{R}$ with $\left | x \right | \leq 1$: $$\left | \exp(x) – 1 \right | \leq \left | x \right | \cdot (e-1)$$ and $\exp(0) = 1$ …………………………………………………………………………………………………………… In order to proof that the […]

Find all continuous functions $f(x)^2=x^2$

Find all functions $f$ which are continuous on $\mathbb R$ and which satisfy the equation $f(x)^2=x^2$ for all $x \in \mathbb R$. Clearly $f(x)=x, -x, |x|, -|x|$ all satisfy the condition. However, how can I show that these must be the only possible choices? The condition guarantees that $|f(x)|=|x|$, for all $x$ so I think […]

exponential matrix

Hi i am trying to understand the exponential matrix: When is exponential matrix function $e^{At}$ integrable where A is an $n \times n$ matrix and $t$ is an $n$-dimensional vector? By integrable i mean indefinite integral over $(0, \infty)$ Do we have to say something about the norm of the matrix? Can we say the […]

Does $d(x+u, y + v) \le d(x, y) + d(u,v)$ holds for every metric?

The title said it, I want to prove that $$ d(x+u, y + v) \le d(x, y) + d(u,v) $$ for every metric $d$. If the metric is induced by a norm, i.e. $d(x,y) := ||x-y||$, then this is easy. \begin{align*} d(x+u, y+v) & = ||x+u – (y+v)|| \\ & = ||x-y + u – […]

Continuous bijections from the open unit disc to itself – existence of fixed points

I’m wondering about the following: Let $f:D \mapsto D$ be a continuous real-valued bijection from the open unit disc $\{(x,y): x^2 + y^2 <1\}$ to itself. Does f necessarily have a fixed point? I am aware that without the bijective property, it is not necessarily true – indeed, I have constructed a counterexample without any […]