# Tagged Questions

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### 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 ...
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### 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? ...
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### 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 ...
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### 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 $\|v\|_1\leq K\|v\|_2$ ...
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### 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 $B$ ...
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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 $\|f\|_p = \left(\displaystyle\sum_{j=0}^{\infty} ... 3answers 1k views ### Example of two norms on same space, non-equivalent, with one dominating the other I am looking for an example (with proof) of two norms defined on the same vector space, such that the norms on the two spaces are NOT equivalent, but such that one norm dominates the other... 1answer 205 views ### Linear isometry between$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$$$c_0 := \{x_n \in l^\infty: \lim x_n = 0\} ... 1answer 876 views ### On the norm of a quotient of a Banach space. Let E be a Banach space and F a closed subspace. It is well known that the quotient space E/F is also a Banach space with respect to the norm$$ \left\Vert x+F\right\Vert_{E/F}=\inf\{\left\Vert ... 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 ... 1answer 140 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 $$\|(x,y)\|_N=N((\|x\|, ... 1answer 212 views ### The Principle of Condensation of Singularities Let X, Y be Banach spaces and \{T_{jk} : j,k \in\Bbb N\} be bounded linear maps from X to Y. Suppose that for each k there exists x\in X such that \sup\{\lVert T_{jk} x\rVert : j ... 1answer 78 views ### How to show that the scalar product on a vector space extends by continuity to a scalar product on the completion of the vector space? I'm trying to solve the following problem: Assume H_0 is a vector space equipped with a scalar product. Complete H_0 with respect to the norm \Vert x \Vert = \langle x,x \rangle^{1/2}. We ... 1answer 240 views ### C^1 [0,1] with different norm If the space C^1 [0,1] is equiped with norm \Vert \cdot\Vert_1,where$$ \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? ... 0answers 313 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 ... 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 ...
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 ...