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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If $f$ is a holomorphic function to a Banach space whose image is contained in a dense subspace, is the image of $f'$ also contained in this subspace?
Let $f$ be a holomorphic function from an open subset of $\mathbb{C}$ to a complex Banach space. In this answer, it is proved that the Cauchy integral formula still holds, and it follows that $f$ is ...
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There are two definition of resolvent set
Recently, when I studied the spectrum of an operator, I came across two definitions of the resolvent set. In Kreyszig’s book, the definition is as follows:
Let $X \ne {0}$ be a complex normed space ...
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Suppose $D^3f$ exists, prove $D_AD_AD_A f$ exists.
Let $A,B,C$ be (banach) normed spaces. On $A\times B$ we consider the supremum norm. Let $X\subseteq A \times B$ be an open set.
Let $f: X \rightarrow C$ and suppose its third differential map exists,...
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Intuition connecting the definition of norm of a number with its p-adic expansion
Let $\alpha\in K$ where $K$ is an algebraic extension of $\mathbb{Q}_p$ and $n:= [K : \mathbb{Q}_p]$.
Let $f(x) : = x^n + a_{n-1}x^{n-1}+...a_0$ be a minimum polynomial of $\alpha$ over $\mathbb{Q}_p$....
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Using self-adjoint operator to define Euclidean norm
I am trying to understand definition of Euclidean norm on a finite dimensional space, $\mathbb{R}^{n}$, denoted as $\mathbb{E}$. The dual space is denoted by $\mathbb{E}^{*}$.
In the attached ...
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A characterization of norms coming from inner products
Let $X$ be a normed vector space and fix two distinct vectors $u$ and $v$ in $X$. Consider the set: $E(u,v) = \{z\in X | |z-u| = |z-v|\}$.
An interesting fact I observed is that if the norm that ...
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Are all norms of complex fields spherical?
Any complex vector space $\mathbb{C}^{n}$ is isomorphic to a real vector space $\mathbb{R}^{2 n}$. I was wondering, however, if converting complex vector spaces to real ones offers more freedom with ...
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Remote generation of functions
The following problem appeared in my current quest for understanding fundamental physics. It is a bit complicated, but I try to explain it as clearly as possible. The problem has to do with the ...
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Equivalency of a mixed Sobolev norm
I was working on a problem and the following question arose. Consider the norm
$$ \| (-\Delta)^\gamma (1-\Delta)^{- \gamma /2} f\|_{L^2}.$$
It appears to combine features of the usual inhomogeneous ...
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Schur's test with weight for infinite matrices and weighted $\ell^2$ spaces
I have come across Schur's test in the presentation here. It states the following.
Let $A=(a_{ij})$ be an infinite complex matrix, $(p_n)$ and $(q_n)$ be two sequences of positive real numbers, and ...
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How good is computing $\left\|f\right\|_p$ with large $p$ as a numerical approximation for $\left\|f\right\|_{\infty}$ for functions $f$?
Let's suppose that I wanted to compute $\left\|f\right\|_{\infty}=\sup_{t \in \mathcal{T}} \left|f(t)\right|$ for a $f$ that may not be easy to optimize. This is the infinity norm of a function, and ...
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Sequence of vector units $x_1, x_2, \ldots$ in a normed space $X$ such that they are all linearly independent and their distances are $>1$.
I'm on my first semester of the Functional Analysis course that we have in my university. I've been stuck on this particular problem our professor presented to us not long ago for a while now:
Let $...
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Motivation for Riesz' Lemma
I have known the Riesz' lemma for a long time. I know it is very important as it help characterise finite dimensional normed linear spaces. But its statement does not seem very natural to me. What can ...
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Uniqueness of extrema for convex functions in a normed space
The statement and the proof are taken from a real analysis book.
Theorem: Let (V, ρ) be a normed space and $A$ be a convex subset of V. Consider
a concave (convex) function $f : A → \mathbb{R}$. Then,...
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For normed vector space, can different norms make sequence to different limits? [duplicate]
A vector space can has different norms. A sequence can converge in a norm, but diverge in another norm. However, can there be:
A vector space $V$
Two norms $\left\| \cdot \right\|_1$, $\left\| \cdot \...
