Tagged Questions

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Geometric conditions equivalent to a set being the unit circle for some norm

Here's the question, as in the textbook (Real Mathematical Analysis, Pugh). The unit ball with respect to a norm $||\, \cdot \,||$ on $\mathbb{R}^2$ is  \{ v \in \mathbb{R}^2 : ||\, v \,|| ...
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Proving the inequality of Cauchy-Schwarz in an Euclidean space. [duplicate]

It says let (G, <.,.>) be an euclidean space. Show that for all x, y belonging to G: modulus<x,y> <= sqrt<x,x> * sqrt<y,y> and in the ...
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Which is the correct definition of stationary point for real-valued functions in Euclidean space?

Given a multivariable real-valued function $f$ whose first partials all exist (but which aren't all continuous) at $p$, it is possible that $f$ is not (totally) differentiable at $p$. But since the ...
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Which is the correct Hessian matrix (the standard matrix of a bilinear form)?

Please note the typo in the first entry: $\frac {\partial^2f} {\partial x_1\partial x_2}$ should instead be $\frac{\partial^2f} {\partial x_1\partial x_1}$. Also, this Hessian matrix need not be ...
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Circle Chord Sequence

This is my first post, so be nice! When I was in my first Geometry class in high school, I asked the teacher the following: Given a circle of radius 2a, find the length of the chord running parallel ...
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Which books can I study (learn) better the topics about convergence in $\Bbb R^{n}$ and euclidean space?

I have question. Which books can I study (learn) better the topics about convergence in $\Bbb R^{n}$ and euclidean space ? Please can you give me an advice some book names? Thank you!
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prove that the sphere with a hair in $IR^{3}$ is not locally Euclidean at q. Hence it cannot be a topological manifold.

A fundamental theorem of topology, the theorem on invariance of dimension, states that if two nonempty open sets $U ⊂ R_{n}$ and $V ⊂ R_{m}$ are homeomorphic, then n = m. prove that the sphere with a ...
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Please give an example in $\mathbb{R}^{n}$ of a set that satisfies the upper bound in the Kuratowski Closure-Complement Theorem.

Please give an example in $\mathbb{R}^{n}$ of a set $T$ that satisfies the upper bound of $14$ in the Kuratowski Closure-Complement Theorem. Thanks!
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Orthogonality between three vectors whose coordinates are nondecreasing

Say that a vector $x=(x_1,x_2, \ldots ,x_n)\in {\mathbb R}^n$ is nondecreasing if $x_1 \leq x_2\leq \ldots \leq x_n$. Can anyone show or find a counterexample to the following : if three nondecreasing ...
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Proof by vectors segments, Euclidean.

Hi I have a midterm this Friday, and our prof gave us this practice midterm, I got $6$ of the $10$ questions but couldn't do the last $4$, could you please help me out in this question, I have no idea ...