Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, and integration through the fundamental theorem of calculus.

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What is the relation between the matrix of a bounded linear operator and that of its adjoint?

Let $X$ and $Y$ be finite-dimensional normed spaces, both real or both complex, and let $\dim X = n$ and $\dim Y = m$. Let $E \colon= ( e_1, \ldots, e_n )$ be an ordered basis for $X$, and let $F \...
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Prove this series does not uniformly converge

$$\sum_{n=1}^{\infty}n^2 \sin \frac{x}{n^4}$$ It is easy to show that it absolutely converges. But what about uniform convergence? With M-test: $$|| f_n|| = \sup (| n^2 \sin \frac{x}{n^4}|) \leq \...
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Real Analysis, Folland Corollary 2.9

This follows from propostion 2.7: If $\{f_j\}$ is a sequence of $\overline{\mathbb{R}}$-valued measurable functions on $(X,M)$, then the functions $$\begin{aligned} g_1(x) = \sup_{j}f_j(x), \ \ \ \ ...
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How do you prove that $\{ Ax \mid x \geq 0 \}$ is closed?

Let $A$ be a real $m \times n$ matrix. How do you prove that $\{ Ax \mid x \geq 0, x \in \mathbb R^n \}$ is closed (as in, contains all its limit points)? The inequality $x \geq 0$ is ...
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Arithmetic implications of different ways to geometrically construct an Hilbert's curve

I have a question on the relation between the geometric and the arithmetic representation of the Hilbert's space-filling curve. Geometric representation: consider the Hilbert's curve $f_h:[0,1]\...
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1answer
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Drawing large rectangle under concave curve

Let $f$ be a continuous concave function on $[0,1]$ with $f(1)=0$ and $f(0)=1$. Does there exist a constant $k$ for which we can always draw a rectangle with area at least $k\cdot \int_0^1f(x)dx$, ...
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Theorem regarding Change of Variables in finite dimesnion

My question is based on Change of Variables in Multiple Integrals II Peter D. Lax > It is not necessary to read the paper before answering this question.The author tried to prove change of variables ...
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$f \in C^2(\mathbb R)$ , $(f(x))^2 \le 1$ ; $(f'(x))^2+(f''(x))^2 \le 1 $ ; then is $(f(x))^2+(f'(x))^2 \le 1 $?

Let $f \in C^2(\mathbb R)$ be such that $$(f(x))^2 \le 1 ; (f'(x))^2+(f''(x))^2 \le 1 , \forall x \in \mathbb R$$ Then is it true that $(f(x))^2+(f'(x))^2 \le 1 , \forall x \in \mathbb R$ ? I ...
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Differentiablity at $0$ of a function $f: \mathbb R \to \mathbb R$ which is twice differentiable in $\mathbb R \setminus \{0\}$

Let $f: \mathbb R \to \mathbb R$ be a function , twice differentiable in $\mathbb R \setminus \{0\}$ such that $f'(x)<0<f''(x) , \forall x <0$ and $f'(x)>0>f''(x) , \forall x >0$ ; ...