Questions tagged [normed-spaces]

Question 1

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,...

Question 2

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 $\mathbb{Q}_p$....

Question 3

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 ...

Question 4

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 ...

Question 5

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 ...

Question 6

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 ...

Question 7

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 ...

Question 8

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 ...

Question 9

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 ...

Question 10

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 $...

Question 11

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 ...

Question 12

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,...

Question 13

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 \...

Question 14

Given a matrix $A$, the norm $\|A\|_{\infty\rightarrow 1}$ is $\max_{x:\|x\|_{\infty}=1}\|Ax\|_1$. A reference says that $$\|A\|_{\infty\rightarrow 1}=\max \sum_{i,j}A_{ij}c_id_j,$$ where the maximum ...

Question 15

It is know that if $f:\mathbb{R}^n\to \mathbb{R}$ is differentiable and $[a,b]$ is the segment from $a$ to $b$, there is at least one number $c\in [a,b]$ such that $df(c)(b-a)=f(b)-f(a)$. The proof ...

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Questions tagged [normed-spaces]

Suppose $D^3f$ exists, prove $D_AD_AD_A f$ exists.

Intuition connecting the definition of norm of a number with its p-adic expansion

Using self-adjoint operator to define Euclidean norm

A characterization of norms coming from inner products

Are all norms of complex fields spherical?

Remote generation of functions

Equivalency of a mixed Sobolev norm

Schur's test with weight for infinite matrices and weighted $\ell^2$ spaces

How good is computing $\left\|f\right\|_p$ with large $p$ as a numerical approximation for $\left\|f\right\|_{\infty}$ for functions $f$?

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 $>1$.

Motivation for Riesz' Lemma

Uniqueness of extrema for convex functions in a normed space

For normed vector space, can different norms make sequence to different limits? [duplicate]

Question about an expression of the norm $\|A\|_{\infty\rightarrow 1}$

Does the Mean Value Theorem work for differentiable functions $f:E\to \mathbb{R}$ defined on any normed space $E$?

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