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 […]

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!

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, […]

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.

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 $\alpha \in \Bbb{Q} \cap (0,1)$: […]

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 […]

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 […]

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 […]

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 […]

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? […]

Intereting Posts

Construction of a right triangle
Vertical bar sign in Discrete mathematics
Property of an operator in a finite-dimensional vector space $V$ over $R$
Closing up the elementary functions under integration
Qual question archives?
The center of the dihedral group
If $\lambda_n \sim \mu_n$, is it true that $\sum \exp(-\lambda_n x) \sim \sum \exp(-\mu_n x)$ as $x \to 0$?
Axiomatization of $\mathbb{Z}$
Does $\pi$ contain infinitely many “zeros” in its decimal expansion?
Equivalences of continuity, sequential convergence iff limit (S.A. pp 106 t4.2.3, 110 t4.3.2)
Dimension of the solution of a second order homogenous ODE
$(A\cap B)\cup C = A \cap (B\cup C)$ if and only if $C \subset A$
A list of books for discovering mathematics using computer software
What is the relation between connections on principal bundles and connections on vector bundles?
How many non isomorphic groups of order 30 are there?