Questions tagged [normed-spaces]

Question 1

Let $\{H_i\}_{i\in I}$ be a directed system of Hilbert spaces over a poset $I$ such that $H_i\subset H_j$ whenever $i\leq j$. We consider the direct limit $H=\cup_{i\in I} H_i$ and equip it with the ...

Question 2

Consider a self-concordant function $f(\cdot)$. This implies; $$ \|f'''(x)[u]\| \leq M \|u\|_{f''(x)}^{3/2} \; \text{for some M and} \; \forall u \in \mathbb{R}^{n}, \text{and} \; x \in \text{domain} \...

Question 3

Let $V$ be a normed vector space, $D\subseteq V$ and $\{f_n\}_{n\in\mathbb{N}}$ a sequence in $V^*$ such that for all $x\in D$, the sequence $\{f_n(x)\}_{n\in\mathbb{N}}$ is bounded. Is it true that ...

Question 4

I am studying functional analysis using "Function Analysis An Introduction" by Yuli Eidelman, Vitali Milman, Antonis Tsolomitis. In page $44$ the following exercise (number $24$) is left for ...

Question 5

Let $X$ be a normed vector space, we have $\{x \in X, ||x|| \le 1\}$ is compact iff $\{x \in X, ||x|| = 1\}$ is compact One direction is clear (a closed subset of a compact set is compact), how to ...

Question 6

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 ...

Question 7

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 ...

Question 8

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 9

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 10

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 11

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 12

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 13

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 14

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 15

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 ...

Stack Exchange Network

Questions tagged [normed-spaces]

Understanding bounded sets locally convex direct/inductive limit of topological vector spaces.

Properties of self-concordant functions

Does pointwise boundedness of continuous linear form in a subset of a normed space implies pointwise boundedness in the clousure? [duplicate]

For $f\in E^{\ast}$ If $\ker \left(f \right) \leq E$ is closed then $f$ is bounded. [duplicate]

Compactness of unit ball and sphere

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?

There are two definition of resolvent set

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

Hot Network Questions