Articles of analysis

Showing that the norm of the canonical projection $X\to X/M$ is $1$

How do I show that given $M$ a closed subspace of a normed space $X$, and let $\pi$ be the canonical projection of X onto $X/M$. Prove that $\|\pi\| = 1$. I figure I could use Riesz’ lemma and set $\|\pi\| = 1$, that’s as far as I got. I could also use the fact […]

Convolution of compactly supported function with a locally integrable function is continuous?

Can someone show me the proof that the convolution of a compactly supported real valued function on $\mathbb{R}$ with a locally integrable function is also continuous? I feel that this is a standard analysis result, but cannot remember how to prove it. Thanks!

How to solve this derivative of f proof?

A function $f$ satisfies: $$f”(x) + f'(x)g(x) – f(x) = 0$$ for some function $g$. Prove that if $f$ is $0$ at two points, then $f$ is $0$ on the interval between them. Can someone verify my proof? Scratchwork: So let $I = [a, b]$ and $f(a) = f(b) = 0$. $g(x)$ is some function, […]

Prove or disprove that if $f$ is continuous function and $A$ is closed, then $\,f$ is closed.

If $f:X\to Y$ is a continuous function and $A\subset X$ is a closed set, then $f[A]$ is also closed. I know that if $f[A]$ is closed implies $A$ is closed then $f$ is continuous, but I’m not sure that the converse is true, I can’t find a counterexample either.

Find $\lim (a_{n+1}^\alpha-a_n^\alpha)$

Assume $\alpha \in (0,1)$, and $\{a_n\}$ is a strictly monotone increasing positive series. and $\{a_{n+1}-a_n\}$ is bounded. Find $$\lim_{n \rightarrow \infty}(a_{n+1}^\alpha – a_{n}^\alpha)$$. My idea is first proving for rational numbers , then use a rational sequences to approximate real numbers. But I can only prove for rational numbers. If $\alpha \in \Bbb{Q} \cap (0,1)$: […]

Differentiability-Related Condition that Implies Continuity

I previously asked a related question here that I did not phrase as I intended. This is a revision of that question: It is a well-known fact that differentiability implies continuity. And, for example, the existence and continuity of first partial derivatives implies continuity; but it also implies differentiability. My question is this: is there […]

Number of Pythagorean Triples under a given Quantity

Consider the function $Pt(n)$. It tells us how many primitive Pythagorean Triples there are (below $n$) when any argument $n \in \mathbb{N}$ is plugged in. Is there an ‘exact formula’; i.e. an elementary function of even a combination of known special functions like the Gamma and Error Function, that describes $Pt(n)$ ? Max Edit: I’m […]

Uniform convergence of the Bergman kernel's orthonormal basis representation on compact subsets

Consider the Bergman kernel $K_\Omega$ associated to a domain $\Omega \subseteq \mathbb C^n$. By the reproducing property, it is easy to show that $$K_\Omega(z,\zeta) = \sum_{n=1}^\infty \varphi_k(z) \overline{\varphi_k(\zeta)},\qquad(z,\zeta\in\Omega)$$ where $\{\varphi_k\}_{k=1}^\infty$ is any orthonormal basis of the Bergman space $A^2(\Omega)$ of Lebesgue square-integrable holomorphic functions on $\Omega$. This series representation converges at least pointwise, since the […]

Prob. 23, Chap. 4 in Baby Rudin: Every convex function is continuous and every increasing convex function of a convex function is convex

Here is Prob. 23, Chap. 4 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: A real-valued function $f$ defined in $(a, b)$ is said to be convex if $$ f \left( \lambda x + (1- \lambda) y \right) \leq \lambda f(x) + (1-\lambda) f(y)$$ whenever $a < x < b$, $a […]

Can a non-zero vector have zero image under every linear functional?

Let $X$ be an infinite-dimensional vector space, and let $x_0$ be an element of $X$ such that $f(x_0)=0$ for every linear functional $f$ defined on $X$. Then can we prove that $x_0$ is the zero vector in $X$? Or, is it possible to find a non-zero $x_0$ whose image under every linear functional is zero? […]