# Questions tagged [normed-spaces]

A vector space $E$, generally over the field $\mathbb R$ or $\mathbb C$ with a map $\lVert \cdot\rVert\colon E\to \mathbb R_+$ satisfying some conditions.

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### Norm on the space of vector fields

Let $M$ be a finite dimensional manifold. There exists the notion of the norm of a tangent vector field on M? In other words, the space of tangent vector fields is a normed space? Thank you in advance ...
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### Spanning set of support functionals in dual space

I am currently studying about supporting hyperplane (or, support functional) in dual space. Since, I am new in these topics I met with the following queries: Let $X$ be a normed space and $X^*$ be the ...
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### The space $C_c$ of real-valued compactly supported, continuous functions is not a Banach space under any norm

This answer showed the space $c_{00}$ of compactly supported sequences, is not a Banach space under any norm. I wonder if the same is true for the space $C_c$ of real-valued compactly supported, ...
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### (Copy) Set of linear functionals span the dual space iff intersection of their kernels is {0} .

I have fully understood the following question and got a motivation from it. Set of linear functionals span the dual space iff intersection of their kernels is $\{0\}$. My question is what will be the ...
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### Why is positive definite defined this way?

Norm: A norm on a vector space $V$ is a function $\| \cdot \| : V \to \mathbb R$ which assigns each vector $x$ its length $\|x\| \in \mathbb R,$ such that for all $\lambda \in R$ and $x, y \in V$ the ...
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### Let $X$ be a reflexive space and let $f \in X^*$, $Px = x - \frac{f(x)}{\|f\|} y, \quad x \in X.$

Let $X$ be a reflexive space and let $f \in X^*$. (a) Show that there exists $y \in X$ such that $x - \frac{f(x)}{\|f\|} y \in \ker f$ for every $x \in X$. (b) Let $P : X \to X$ be the mapping defined ...
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### Let $(X, \|\cdot\|)$ be a normed space, $a_1,a_2,\ldots, a_n \in \mathbb{C}$ and $x_1,x_2,\ldots, x_n$ linearly independent vectors of the space $X$.

Let $(X, \|\cdot\|)$ be a normed space, $a_1, a_2, \ldots, a_n \in \mathbb{C}$ and $x_1, x_2, \ldots, x_n$ linearly independent vectors of the space $X$. Show that the following two statements are ...
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### linear vector addition in norm of vectors suggest a line segment in the unit circle

Let $V$ be a normed vector space, which contains two linearlly-independant vectors $x,y$ such that $\|x+y\|=\|x\|+\|y\|$. Prove that the unit circle $\{x \in V:\|x\|=1\}$ contains a line segment. From ...
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### Bound on spherical harmonic L2 norm from L_inf norm

The online notes here state the following as proposition 6.0.1. If $f \in H_d$ then $$\sup_{x \in S^n} |f(x)| \leq \sqrt{\frac{dim H_d}{ \text{vol}(S^{n-1})}} |f|_{L^2},$$ where $H_d$ is the space of ...
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### Reference for isomorphism theorem for Banach spaces

Let $X,Y$ be Banach spaces, and a linear continuous operator $T:X \rightarrow Y$ then we have that, denote by $N =\text{ker }T$ and by $\text{Ran }T$ the closed range of T then,  X / N \cong \text{...
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### rational complex minmax

As the title says, I wanted to understand the following part of a proof in Guttel (Corollary 2, page 6). $f$ is an analytic function and $r_m \in \mathcal{P}_{m-1}/q_{m-1}$ is a rational function with ...
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### Does this relation between Wasserstein distances hold: $W_1(\mu,\nu)\leq W_2(\mu,\nu) \leq ... \leq W_\infty(\mu,\nu)$?

I stumbled upon this interesting statement in this paper: "One interesting observation is that the Wasserstein ambiguity set with the Wasserstein order p = 2 is less conservative, because the 2-...
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### Is there a notation for normed spaces $(X, \| \cdot \|_X)$ and $(Y, \| \cdot \|_Y )$ such that $X = Y$ and $\| x \|_X = \| x \|_Y$ for all $x \in X$?
Suppose that we are dealing with two abstract nomed spaces $(X, \| \cdot \|_X )$ and $(Y, \| \cdot \|_Y)$ such that $X = Y$ (that is, every element of $X$ belongs to $Y$ and every element of $Y$ ...