0
votes
2answers
22 views

Bounding the distance between $L_\infty$ and $L_2$ for a continuous function

Consider a set of continuous (or even differentiable) functions $f_i(x)$, all defined for $x\in [a,b]$ for $i=1\ldots,N$. Can one define a uniform constant $c$ (which may depend on $f$) such that ...
0
votes
1answer
42 views

Frechet derivative of the sup norm function on $C[0,1]$

I know that the derivative in the title does not exist for any x but I do not have a clue why. Could someone explain why this derivative DNE at 0 and I can figure out why it does not exist at x? ...
0
votes
0answers
58 views

Completeness is not preserved under homeomorphism

I am trying to give a counterexample which will make it clear that homeomorphism does not preserve completeness. I have an easy one in mind ($(0,1)$ and $\mathbb{R}$) but I have just thought that ...
0
votes
1answer
83 views

Normed linear space with two norms that are not equivalent, one is complete, what about the other?

I have been searching for an answer to the following question: Given a normed linear space $V$ and two norms that are not equivalent, but $\exists K\in\mathbf{R}$ such that $\|v\|_1\leq K\|v\|_2$ ...
0
votes
0answers
43 views

Largest/smallest cross-norm: A simple question about cross-norms on tensor products of Banach spaces.

This is a very simple dumb question as I’m completely new to the topic. I was reading Wikipedia’s entry on “Topological Tensor Products”, and there’s one thing I’m confused about. Let $ A $ and $ B $ ...
1
vote
1answer
38 views

Proof check for completeness

I'd like to see if the proof I have is adequate. Statement. Let $X$ and $Y$ be Banach space, the product $X\times Y$ is a vector space under coordinate operations with norm $$ \|(x,y)\| = \|x\|_X ...
5
votes
0answers
108 views

renorm a Banach space to make an operator have spectral radius equal to norm

Let $X$ be an infinite-dimensional complex Banach space equipped with the norm $\lVert\cdot\rVert$, and let $T\in\mathcal{L}(X)$ a bounded linear operator on $X$. Let $r(T)$ denote the spectral ...
2
votes
1answer
36 views

Showing that a map defined using the dual is a bounded linear operator from X' into X'

I have trouble answering the second part of the following exercise. Any help would be appreciated! Let $(X, \| \cdot \|)$ be a reflexive Banach space. Let $\{ T_n \}_{n = 1}^\infty$ be a sequence of ...
0
votes
0answers
34 views

Injective norm on tensor algebra of a finite-dimensional Banach space

Suppose that $V$ is a finite n-dimensional Banach space, and suppose that $T(V)$ is the tensor algebra on it. Furthermore, suppose that $T^{(n)}(V)$ is the "abridged" tensor algebra obtained by taking ...
0
votes
1answer
40 views

How to detect reflexivity of the closure

Consider the space of continuous bounded functions on a bounded interval. Its closure for the Lebesgue $L_p$ norm is reflexive when $1 < p < \infty$, but it is not reflexive for $p = 1$. How ...
1
vote
1answer
53 views

norm of Frechet derivative in point.

Let $ E = \mathcal B([0,1], \Bbb R) $ with supremum norm. Now I can define function $ F:E \ni f \rightarrow ||f||^2 - f(0) \in \Bbb R$ My task: 1)Show the differentiability of $F$ in: $ f_0: ...
1
vote
1answer
52 views

Can a vector space be complete for two non-compatible norms?

Let $V$ be a vector space and suppose that we have two non-compatible norms on it, i.e. I distinguish $E = (V, \|\cdot\|_1)$ from $F = (V, \|\cdot\|_2)$ and I ask that $\not\exists C>0 \; ...
0
votes
1answer
62 views

Prove that $|x(a)| + \max \{|x'(t)|: t \in [a,b]\}$ is a norm for a complete space

Prove that the space $C^1([a,b])$ consisting of continuous functions in $[a, b]$ with the norm $|x(a)| + \max \{|x'(t)|: t \in [a,b]\}$ is a Banach space. I can't prove the completeness of this ...
2
votes
3answers
91 views

Norms that $C([0,1])$ to be an incomplete normed space.

I searched all of norms that $C([0,1])$ to be incomplete normed space. But I found only $\|.\|_p$ (for every $1\leq p<\infty$). Are you know another norm on $C([0,1])$ that $C([0,1])$ to be ...
3
votes
1answer
168 views

Closed-form expressions for dual norms of real normed vector spaces

Say that $V$ is a finite-dimensional real normed vector space, where for some $v \in V$ the norm is notated by $\|v\|$. Then say that $V^*$ is the dual space of linear functionals on $V$. The "dual ...
3
votes
0answers
97 views

Proving norm equivalence in $W^{1-1/p,p}(\Omega)$

Define for $p\in [1,\infty)$ and $\Omega=(0,1)^N\subset\mathbb{R}^N$ $$W^{1-1/p,p}(\Omega)=\left\{u\in L^p(\Omega): \ \int_\Omega\int_\Omega\frac{|u(x)-u(y)|^p}{|x-y|^{N-1+p}}dxdy<\infty\right\}$$ ...
0
votes
1answer
80 views

What is a decreasing scale of Banach spaces?

I am a having a hard time understanding a part of an article I am reading. The screen-cap is below. Basically, it's the line labeled (6) that I do not understand. I am not familiar with the circular ...
1
vote
0answers
70 views

Distance of a point from a subspace vs. diameter

Let X = $(\Bbb R^N, \|\cdot\|)$ be a Banach space. Let $x_0 \in S^{N-1} = \{x \in \Bbb R^N : \sqrt{x_1^2+...+x_n^2}=1\}$. Denote $B^N_2 = \{x \in \Bbb R^N : \sqrt{x_1^2+...+x_N^2} \le 1 \}$. Define ...
2
votes
1answer
78 views

Calculating the norm of $S\otimes T$ for bounded linear $S,T$.

Let $X,Y,W,Z$ be Banach Spaces. Let $X\otimes_{\pi}Y$ denote the tensor product endowed with the projective norm. If I have $S\in B(X,Z)$, $T\in B(Y,W)$, it is straightforward to show that $S\otimes ...
5
votes
3answers
192 views

Sequences are Cauchy depending on the norm?

Assume that we have two Banach spaces $B_1, B_2$ with their underlying sets being identical. Is it possible that a Cauchy sequence in one of the spaces would fail to be Cauchy in the other? I am ...
3
votes
3answers
722 views

equivalent norms in Banach spaces of infinite dimension

Suppose $ X $ is a Banach space with respect to two different norms, $ \|\cdot\|_1 \mathrm{ e } \|\cdot\|_2 $. Suppose there is a constant $ K > 0 $ such that $$ \forall x \in X, \|x\|_1 \leq ...
1
vote
2answers
139 views

Proof of non-strictly convexity of $l_1$ and $l_{\infty}$

Given the Banach space $l_p = \left\{f\in R^n : \|f\| \leq \infty\right\}$ for $1\leq p \leq \infty$, we define the following norms $\|f\|_p = \left(\displaystyle\sum_{j=0}^{\infty} ...
3
votes
3answers
1k views

Example of two norms on same space, non-equivalent, with one dominating the other

I am looking for an example (with proof) of two norms defined on the same vector space, such that the norms on the two spaces are NOT equivalent, but such that one norm dominates the other...
4
votes
1answer
208 views

Linear isometry between $c_0$ and $c$

The following question is an exercise and so I'm just looking for advices and not for answers if it's possible. I have the following sets in $l^\infty$ $$c_0 := \{x_n \in l^\infty: \lim x_n = 0\} ...
14
votes
1answer
916 views

On the norm of a quotient of a Banach space.

Let $E$ be a Banach space and $F$ a closed subspace. It is well known that the quotient space $E/F$ is also a Banach space with respect to the norm $$ \left\Vert x+F\right\Vert_{E/F}=\inf\{\left\Vert ...
0
votes
1answer
34 views

A limit superior question in the context of the Neumann series

I'm trying to understand a step in the proof that the Neumann series converges: Let $X$ be a Banach space and $T\in L(X)$ (the space of bounded, i.e. continuous linear maps $X\to X$). It is known ...
3
votes
1answer
141 views

Absolute norms and 1-unconditional sums

Absolute norm Let $X$ and $Y$ be Banach spaces. Let $Z=X\times Y$ a norm $\|\cdot\|_N$ on $Z$ is called absolute if there is a function $N\colon R^2\rightarrow R$ such that $$ \|(x,y)\|_N=N((\|x\|, ...
2
votes
1answer
217 views

The Principle of Condensation of Singularities

Let $X$, $Y$ be Banach spaces and $\{T_{jk} : j,k \in\Bbb N\}$ be bounded linear maps from $X$ to $Y$. Suppose that for each $k$ there exists $x\in X$ such that $\sup\{\lVert T_{jk} x\rVert : j ...
1
vote
1answer
79 views

How to show that the scalar product on a vector space extends by continuity to a scalar product on the completion of the vector space?

I'm trying to solve the following problem: Assume $H_0$ is a vector space equipped with a scalar product. Complete $H_0$ with respect to the norm $\Vert x \Vert = \langle x,x \rangle^{1/2}$. We ...
2
votes
1answer
243 views

$C^1 [0,1]$ with different norm

If the space $C^1 [0,1]$ is equiped with norm $\Vert \cdot\Vert_1$,where $$ \Vert f\Vert_1=|f(0)|+\Vert f'\Vert _{C}=|f(0)|+\sup_{t\in [0,1]}|f'(t)| $$ for any $f\in C^1 [0,1]$, is this space Banach? ...
2
votes
0answers
319 views

Bound on inverse operator

Define $X = {C^{2, \alpha}}(U \times [0,T])$ and $Y = {C^{0, \alpha}}(U \times [0,T])$ where $U$ is some real interval. Let $F:X \to Y$ be a map. Let $DF(g):X \to Y$ be a bounded linear operator for ...
3
votes
2answers
116 views

Geometric Sums in Banach Algebra

Let $E$ be a Banach Algebra with identity, and $v\in E$, so that $||v|| < 1$. The geometric series $w = \sum_{k=0}^\infty v^k$ converges in the norm. I can show that $||w|| \le ...
7
votes
2answers
296 views

Is a completion of an algebraically closed field with respect to a norm also algebraically closed?

Assume we have an algebraically closed field $F$ with a norm (where $F$ is considered as a vector space over itself), so that $F$ is not complete as a normed space. Let $\overline F$ be its completion ...