# Tagged Questions

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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### If $f$ is defined on $\mathbb{R}$ and $f$ is unbounded, is it necessarily true that $\lim_{|x|\to\infty} |f(x)| = \infty$?

This question comes from the following problem: A real-valued function $f$ defined on $\mathbb{R}$ has the following property: For every positive $\epsilon$, there exists positive $\delta$ such ...
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### Indicator function for limsup, liminf

If $A_i$ is a sequence of sets, define$$\liminf_i = \bigcup_{j = 1}^\infty \bigcap_{i = j}^\infty A_i, \quad \limsup_i A_i = \bigcap_{j = 1}^\infty \bigcup_{i = j} A_i.$$Given a set $D$ define the ...
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### Modified Wave Equation: Bound $\int u^2 \, dx$

I'm studying for a qualifying exam and I can't figure this problem out: Suppose $B \subset \mathbb R^n$ is the unit ball centered at the origin and that $u$ is a smooth solution of \begin{align*} ...
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### line integral of 3D vector field

Suppose I have a 3D vector field $\vec v(x,y,z)=(v_1,v_2,v_3)$ and I want to compute $$\int_C \vec{v}\cdot \vec n\, dS$$ where $C$ is the unit circle $C\equiv\{(x,y)\in\mathbb{R}^2\,:\, x^2+y^2=1\}$ ...
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### Show the countability of the non-zero elements when sum of elements is less than $\infty$ on the extended reals.

Show that for $(x_{\alpha})_{\alpha \in A}, x_{\alpha} \in [0, +\infty]$ where $\sum_{\alpha \in A}x_{\alpha} < \infty$ the number of non-zero elements is at most countable. Note $A$ can be ...
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### Norm of gradient of velocity field

If $\mathbf{u}(x,y,z,t)=(u,v,w):\mathbb{R}^3\times[0,+\infty)\to\mathbb{R}^3$ denotes a velocity field, what is the definition for $\|\nabla\mathbf{u}\|_{L^{\infty}}$? I know that $\nabla\mathbf{u}$ ...
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### Is the Fourier transform a unitary isomorphism between $L^2(\mathbb{T}^n)$ and $\ell^2(\mathbb{T}^n)$

I am reading through Folland's "Real Analysis", and it's clear that if $f\in L^2(\mathbb{T}^n)$, then $\{\hat{f}(\kappa)\}\in\ell^2(\mathbb{T}^n)$, and the norms of those two are equal. However, it's ...