# 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.

3,185 questions
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### A Vector Divided by it's Distance to a Subspace Yields a Vector with Distance 1 to Subspace

Let X be a normed vector space, S a subspace of X and x∈X. Distance is defined by: $$|x,S|:=\inf||x-s||, x\in X, s\in S$$ How does one prove that for: $$z:=\frac{x}{|x,S|}\Rightarrow |z,S| = 1$$ This ...
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### p - adic norm not equivalent to usual norm!

I know that the two norms: $p$ - adic norm and the usual norm ($\left| \cdot \right|$) defined on $\mathbb{Q}$ are not quivalent. This is clearly because the $p$ adic norms staisfies the strong ...
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### A question regarding Radon-Riesz property

I want to show that if a normed linear space has Radon-Riesz property, then it has the semi Radon-Riesz property. We know that a normed linear space is said to have Radon-Riesz property if for every ...
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### Complement a finite dimensional subspace in a Banach space

Given a Banach space $(X,\|,\|)$, and a finite dimensional subspace $F \subset X$, is it always possible to choose a closed linear complement to $F$. Explicitly, I mean to say, will there always exist ...
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### Check if $||.||_{1}$ and $||.||_{\infty}$ are strictly convex norms on $\mathbb{C}^n$ for $n \geq 2$.

A norm is strictly convex on a given normed space $X$ if, for every $x,y \text{ with } x \neq y \in X$ of norm 1, we have that $||x + y|| < 2$. $||.||_{1} = \sum_{j}^{n}|x_j|$ Say we take the set ...
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### Convergence of trace class operators in Hilbert Schmidt norm

Let $\mathscr{A}_n$ be a sequence of trace-class operators on a Hilbert space $\mathcal{H}$ and let further $\mathscr{A}$ be another trace-class operator on the same space. Assume that $\mathscr{A}_n$...
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### How is “continuous dependence on initial conditions” defined? And how to prove it?

EDIT: I tried to prove the continuous dependence of the problem by somehow use a weak Maximum principle and posted a modified Version of this post on mathoverflow: https://mathoverflow.net/questions/...
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### Counterexample for “continuous image of closed and bounded is closed and bounded” (in normed spaces).

It's well known that: If $X$ is a finite-dimensional normed space, $C$ is a closed and bounded subset of $X$ and $f:C\subset X\to X$ is continuous, then $f(C)$ is closed and bounded. If $X$ is any ...
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### Examine whether a linear map is continuous and determine the operator norm

Let $P$ be the space of all real-valued polynomials, defined on $\mathbb{R}$. For a polynomial $p\in P$, such that $p(t)= \sum_{k=0}^na_kt^k$, set $||p||:= \sum_{k=0}^n|a_k|$. Consider the linear ...
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### Questions about deriving the dual space of $l^{1}$

I am an engineering student and I am reading the book "Introductory Functional Analysis " by kreyszig and am lost in the proof of finding the dual space of the $l^{1}$ space . Here is how author ...
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### Is it possible that $\forall y \in X \quad \|y-b\| < \max(\|y-a\|, \|y+a\|)$?

Let $(X, \|\cdot\|)$ be a normed space. Let $a, b \in X$ be noncollinear vectors. Is it possible that \begin{equation} \forall y \in X \quad \|y-b\| < \max(\|y-a\|, \|y+a\|)? \end{equation} So ...
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### Are the spaces $T_pM$ and $\mathbb R^n$ homeomorphic?

Let $T_pM$ be the tangent space at a point $p$ in a n-dimensional smooth manifold $M$. In addition, if we assume $(M,g)$ as a smooth Riemannian manifold, then $T_pM$ is a n-dimensional real normed-...
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### Why are reflexive spaces called like that?

A normed spaces $(X, \| \cdot \|)$ is called reflexive if the evaluation map $X \to X^{**}$ is an isomorphism. If $X$ is reflexive, it's not analytically distinguishable from it's bidual space $X^{**}$...
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### The strong, weak and weak-star topologies coincide on finite dimensional spaces.

Let $E$ be a finite dimensional normed linear space. I have been able to show using set inclusion that $s=w=w^*$, where $s,\,w$ and $w^*$ represent strong, weak and weak-star topologies, repsectively. ...
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### Differentiability, linear operators

Let $Y$ be a complete normed linear space, and let $M$ denote the space of bounded linear operators from $Y$ to itself. Let $L : M → M$ be the map defined by $L(A) := A^2$. I am supposed to show that ...
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### Alternatives to Gram-Schmidt

So I'm curious if what I'm about to say is well-known and/or true, because I don't really have time to investigate right now. By inspecting the graphs of sets of orthonormal polynomials (like ...
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### Proof of the Alaoglu Theorem

I was reading through the proof of the Alaoglu theorem which states Let $X$ be a normed space Then the unit ball in $X^*=B^*$ is compact with respect to the $weak^*$ topology. The proof goes as ...