# Tagged Questions

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### Finite-dimensional subspace normed vector space is closed

I know that probably this question has already been answered, but I'd like to present my attempt of solution. Let $(E,\|\cdot\|)$ be a normed vector space and let $F\subseteq E$ be a ...
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### Proof the that compact support is a vector space

Currently I am studying for my exam of Real Analysis, however there is one thing that I do not seem to get. Given: $$\mathrm{Supp}(f):=\overline{\{x\in\mathbb{R}^n:f(x)\neq 0\}}$$ the support of ...
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### Extenstion of Intermediate Value Theorem.

Let $f:[0,1]^{d}\longrightarrow \mathbb{R}^{d}$ with $d\geq 2$. $f$ is continuous and let $c\in (0,1)$. If we have that $f(0,...,0)<<(c,...,c)$ and $f(1,...,1)>>(c,...,c)$, is there an ...
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### Let $V$be a vector space. Prove/Disprove: There is a norm $\|\cdot\|$, such that all subsets of $V$ are open sets in $(V,\|\cdot\|)$.

The Assignment: Let $V$ be a vector space over $\mathbb{R}$ with $V \not= \{0\}$. Prove or disprove: There is a norm $\|\cdot\|_d$ on $V$, such that all subsets of $V$ are open sets in ...
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### How to prove this claim?

For $x=(x_1,\dots,x_p)$ define two norms $\|x\|_1=|x_1|+\cdots+|x_p|$ and $\|x\|=\big(\sum_{i=1}^{p}x_{i}^{2}\big)^{\frac{1}{2} }$. Find the largest constant $a>0$ and the smallest constant ...
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### Need help understanding the concept of the Jacobian Matrix and its relation to differentiation

As the question suggests, I need help understanding the concepts around the following differentiation of vector-valued functions: 1) The Norm. I understand that Norm to be defined as follows: In the ...
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### Is an infinite linearly independent subset of $\Bbb R$ dense?

Suppose $(a_n)$ is a real sequence and $A:=\{a_n \mid n\in \Bbb N \}$ has an infinite linearly independent subset (with respect to field $\Bbb Q$). Is $A$ dense in $\Bbb R?$
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### Prove that a given subspace of $C[-1,1]$ with $L^2$ norm is closed

Let $H= C[-1,1]$ with $L^2$ norm and consider $G=\{f \in H \mid f(1) = 0\}$. Show that $G$ is a closed subspace of $H$. I've been trying to prove this for a while but i can't establish that given ...
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### Find an inner product on $C[0,1]$ such that {$2^{1/2}\sin(n\pi x):n\ge 1$} is an orthonormal set

Find an inner product on $C[0,1]$ such that {$2^{1/2}\sin(n\pi x):n\ge 1$} is an orthonormal set and verify that this set is orthonormal for your choice of inner product. I'm pretty much stumped for ...
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### Prove that a compact subset of a normed vector space is complete

Prove that a compact subset of a normed vector space is complete . My thought process: Suppose S is a compact subset of $(V,\parallel . \parallel)$. Since S is compact, it is closed and bounded, so ...
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### Strictly convex absolutly 1 homogeneous function

Is every strictly convex, 1-homogeneous function on $\mathbb R^d$ simply a multiple of the Euclidean norm? Update: The above is no, since any p-norm on $\mathbb R^d$ is strictly convex and ...
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### Prove that if $L(S)$ is bounded, where $S$ is the unit sphere of $U$, then $L$ is Lipschitz.

Let $U$ and $V$ be normed linear spaces over $\mathbb{R}$, and $L : U \mapsto V$ a linear function. Prove that if $L(S)$ is bounded, where $S$ is the unit sphere of $U$, then $L$ is Lipschitz. There ...
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### $\ell_{p}$ space closed to addition

I'm trying to show that $\ell_{p}$ is a vector space for any $1\leqslant p<\infty$ . So given two infinite series $\left(x_{n}\right)_{n=1}^{\infty}$ and $\left(y_{n}\right)_{n=1}^{\infty}$ ...
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### Differentiation continuous iff domain is finite dimensional

Let $A\subset C([0,1])$ a closed linear subspace with respect to the usual supremum norm satisfying $A\subset C^1([0,1])$. Is $D\colon A\rightarrow C([0,1]), \ f\rightarrow f'$ continuous iff $A$ ...
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### Intersection of $|z_1 - x|=r$ and $|z_2 - y|=r$

Let $x,y \in \mathbb{R}^k$ ($k\geq 3$), $|x-y|=d>0$ and $r>0$. Prove that if $2r>d$, then there are infinitely many $z\in \mathbb{R}^k$ such that $|z-x|=|z-y|=r$. Here's what I have ...
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### Boundedness of Surfaces in $\mathbb R^3$

GIven an equation such as $ax^2+by^2+cz^2+dxy+exz+fyz=g$ where $a,b,c,d,e,f,g\in \mathbb R$, How can we tell if the surface described is a bounded one without explicitly plotting a graph?
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### Is this class of periodic functions closed under the (circular) convolution operation ? Help in proving.

I am studying the properties of a particular class of functions, and I'd appreciate some help in proving a property of that class. I started with a class of functions and made some modifications to ...
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### Condition on $c$ for a contraction map

I am extending Example 2.2 on this sheet. Suppose $f(x(s),s)$ is such that $|f(x(s),s)-f(y(s),s)|\leq K |x-y|$ for some $K>0 ---(1)$ and $x,y\in C[0,t_f]:\,\,\,t_f<\infty$ Also, let $T$ ...
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### Field for bounded real sequences

On page 12 of Luenberger's Optimization by Vector Space Methods, he claims that the set of bounded real sequences is vector space with addition defined component-wise. Ok, but what is the underlying ...
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### How to prove $C_1 \|x\|_\infty \leq \|x\| \leq C_2 \|x\|_\infty$?

I want to prove the following theorem (no idea whether it has a name): Let $V = \mathbb{R}^n$ or $\mathbb{C}^n$ and $\|\cdot\|$ be a norm on $V$. Then, there exist $C_1, C_2 > 0$ such that for all ...