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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5
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2answers
3k views

Is a norm a continuous function?

Is a norm on a set a continuous function with respect to the topology induced by the norm? Is a topology on the set that can make the norm continuous (i.e. the topology that is compatible with the ...
2
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1answer
196 views

Compute an operator norm

Consider the operator $M$ acting on the space $\mathbb{R}[X]$ of real polynomials by $Mp(x)=xp(x)$. We equip $\mathbb R[X]$ with the $L^2$ norm $$ \|p\|^2=\int p(x)^2d\mu(x), $$ where $\mu$ is a ...
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1answer
221 views

To construct a counterexample of normed space

Please construct a counterexample for the following: $A$ is normed space and $M$ is a dense subspace of $A$, if there is a functional $f$ such that $f(M) = 0$, then $f=0$. Besides, if $A$ is a Banach ...
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vote
0answers
150 views

Neumann series in an incomplete normed algebra

Let $\mathcal{A} \equiv (A, \|\cdot\|_A)$ be a unital (associative) normed algebra over the real or complex field, and assume that $\mathcal{A}$ is not complete. Provided $\mathcal{B}_\mathcal{A}$ is ...
2
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1answer
128 views

Completing a normed space

Is the completion of $\{x=(x_n)|x_n\in \mathbb R \text{ and  for a given } x,\text{ only finitely many } x_n\neq0\}$ equipped with the norm $\|x\|:= |x_1|+|x_2|+...$ simply the set of all real ...
3
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1answer
814 views

Show that the norm of the multiplication operator $M_f$ on $L^2[0,1]$ is $\|f\|_\infty$

I'm having some (hopefully small) issues computing the norm of an operator. Firstly, the problem, For $f\in L^\infty[0,1]$, define $M_f: L^2[0,1]\to L^2[0,1]$ by $M_f(g)(x) = f(x)g(x)$. Show that ...
0
votes
2answers
171 views

Why does $\lVert L(x) \rVert \leq \lVert L \rVert\,\lVert x \rVert$?

Why does $\lVert L(x) \rVert \leq \lVert L \rVert\,\lVert x \rVert$? If $L$ is a linear map between Banach spaces $V$ and $W$, why is this true? Also, is this true for $L$ not a linear map? Thanks! ...
2
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0answers
51 views

Embedding of $l^p$ space [duplicate]

Possible Duplicate: Inequality between $\ell^p$-norms First, I'll start by simply showing you the problem: Let $1 \leq p \leq q < \infty$. Determine that $l^p \subseteq l^p$ by proving ...
1
vote
1answer
110 views

Dense subset of given space

If $E$ is a Banach space, $A$ is a subset such that $$A^{\perp}:= \{T \in E^{\ast}: T(A)=0\}=0,$$ then $$\overline{A} = E.$$ I don't why this is true. Does $E$ has to be Banach? Thanks
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0answers
45 views

What are these real-valued functions called?

Let $V\;$ be some normed $\mathbf{R}$-vector space, and $n > 0$ an integer. First, What is the name of the the class of mappings $\prod_i f_i(x_i):V\;^n\to\mathbf{R}$ for some ...
2
votes
1answer
87 views

Linear forms of $L^1$

I am an engineering student who has unwittingly taken a module in functional anylysis which, unfortunately, is ever so slightly over my head. I would greatly appreciate if you could either point me ...
4
votes
1answer
860 views

Are isometric normed linear spaces isomorphic?

I should know the answer to this (and I did some time ago, but have forgotten): If the normed linear spaces $X$ and $Y$ are isometric (there is a bijective map from $X$ to $Y$ that preserves ...
4
votes
1answer
243 views

Shortest path on unit sphere under $\|\cdot\|_\infty$

Let $X$ be $\mathbb{R}^3$ with the sup norm $\|\cdot\|_{\infty}$. Let $Y=\{x\in X: \|x\|_{\infty}=1\}$. For $x,y\in Y$ define $d(x,y)$ to be the arc length of shortest paths on $Y$ joining $x,y$. (It ...
4
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1answer
1k views

Banach space of Lipschitz functions

Let $X$ be a compact metric space, and $F$ the space of all lipschitz functions $X \to \mathbf{C}$. Let $|f|_L$ be the least Lipschitz constant. We endow $F$ with the norm $||f|| = |f|_L + ...
2
votes
1answer
473 views

A question on linear transformation

Let $T$ be a linear transformation on $\mathbb{R}^{4}$ whose standard matrix is $$\left(\begin{array}{rrrr} 1 & -1 & -1 & -1\\ 1 & 1 & 1 & -1\\ 1 & 1 & -1 & 1\\ ...
2
votes
1answer
218 views

The sup-norm of a diagonalizable operator

I want to get familiar with computing sup-norms of diagonalizable operators on $\mathbf{R}^n$. Suppose that I have a diagonalizable linear map $T:\mathbf{R}^n\to \mathbf{R}^n$ and I consider ...
6
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1answer
759 views

Does the p-norm converge to the max-norm in some norm

Does the $p$-norm on $\mathbf{R}^n$ converge to the max-norm on $\mathbf{R}^n$ as elements in the space of real valued continuous functions on $\mathbf{R}^n$ endowed with some norm? More precisely, ...
9
votes
3answers
458 views

Is there an easy example of a vector space which can not be endowed with the structure of a Banach space

Let $V$ be a real vector space. Is there always a norm on $V$ such that $V$ is complete with respect to this norm? If not, is there an easy counterexample?
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vote
1answer
79 views

How do you get pointwise convergence in the context of normed spaces [duplicate]

Possible Duplicate: Norm for pointwise convergence Let $V=C([0,1],\mathbf{R})$ be the vector space of continuous real-valued functions on $[0,1]$. Let $(f_n)$ be a sequence in $V$. Then ...
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1answer
664 views

Cauchy sequence in a normed space

Let $V$ be a real vector space. Suppose that $\Vert \cdot \Vert_1$ and $\Vert \cdot \Vert_2$ are two norms on $V$ which are equivalent. I suspect the following to be true. Let $(x_n)_{n=0}^\infty$ ...
1
vote
1answer
113 views

tensorisation of linear map

Let $X$ be a Banach space and $T \colon \ell^2\rightarrow \ell^2$ be a bounded linear map. Suppose that the linear map $T\otimes Id_ {X}:\ell^2\otimes X\rightarrow \ell^2\otimes X$ which maps $e_i ...
3
votes
1answer
126 views

Do non-commutative algebras with dense commutative subalgebras exist?

Let $A$ be a normed unital algebra. Suppose that $C\subseteq A$ is a commutative subalgebra which is dense in $A$. I ask myself the following question: Under the above assumptions, is $A$ necessarily ...
3
votes
1answer
70 views

Integral of a function taking values in $c_0$

Let $X$ be a Banach space and assume that $a<b$. For a function $f\colon\left [a,b\right]\to X$ we define a generalization of Riemann integral as follows: a point $u\in X$ we call the integral of ...
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votes
2answers
257 views

If you know the convergent sequences, how do you know the open sets?

I have a homework problem which I feel should be simple but is actually surprisingly tricky. This is why I love math sometimes.... Let $X$ be a normed linear space. Suppose $\|\cdot\|_1$ and ...
4
votes
2answers
134 views

Why a norm and not some other function that defines a metric?

If one defines on a $\mathbb{R},\mathbb{C}$-vector space a norm this gives rise to a metric. Why are particularly mappings that satisfy the norm axioms so important that in every book for beginners on ...
11
votes
2answers
842 views

Does every $\mathbb{R},\mathbb{C}$ vector space have a norm?

Is there a canonical way to define on any vector space over $\mathbb{K}=\mathbb{R},\mathbb{C}$ a norm ? (Or, if there isn't, can someone give me an example of a vector space over $\mathbb{K}$ that is ...
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vote
0answers
660 views

Centroid under the Chebyshev distance

I want to find the centroid (point which minimizes the sum of distances) of a set of points in the 2-dimensional plane using the Chebyshev distance ($\textbf{L}_\infty$ norm). I think the answer is ...
2
votes
0answers
257 views

Proof of equivalence in strictly convex spaces

I'm trying to understand a proof and from my current understanding there is one thing not clear/wrong. Maybe you can help: Let $(X, \lVert\cdot\rVert)$ a normed space and $B(0,1)$ the closed unit ...
5
votes
1answer
255 views

Characterization of normed vector spaces of finite dimension

I have this problem: Let $E$ be a normed vector space. $S=\{x\in E : ||x||=1\}$. Show that if $S$ is compact then $\dim E$ is finite. This follows directly from the Riesz's lemma, but in the ...
7
votes
3answers
1k views

Operator norm on product space

I have a bilinear operator $B\colon X \times Y\to Z$ with $X,Y,Z$ normed spaces, and define a norm on $X \times Y$ by $\lVert(x,y)\rVert = \lVert x\rVert_X + \lVert y\rVert_Y$ (using the respective ...
2
votes
2answers
418 views

Point-wise and Norm Convergence of Vectors in a finite dimensional space

I'm trying to prove the theorem, that states, that if I have a normed vector space with a finite dimension (so that each vector I can express as a linear combination $$ ...
18
votes
2answers
4k views

Difference between metric and norm made concrete: The case of Euclid

This is a follow-up question on this one. The answers to my questions made things a lot clearer to me (Thank you for that!), yet there is some point that still bothers me. This time I am making ...
5
votes
1answer
639 views

Different norms on a product space $X \times Y$

It is well known how to define standard product topology on a product space $\prod_{i \in I} X_i$. Assume now that $(X,\lVert \, \cdot \, \rVert_{X})$ and $(Y,\lVert \, \cdot \, \rVert_{Y})$ are ...
4
votes
1answer
235 views

Non-completeness of the space of bounded linear operators

If $X$ and $Y$ are normed spaces I know that the space $B(X,Y)$ of bounded linear functions from $X$ to $Y$, is complete if $Y$ is complete. Is there an example of a pair of normed spaces $X,Y$ s.t. ...