# 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.

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### Prove $\|Tx\|_\infty \le \|x\|$ for all $x$ belonging to a normed space

Here is the question: And here is the given answer: I would like to ask in the first line of the proof, why $\|x^*_k\|$ is less or equal than $1$? I guess zero belongs to $B(x^*_k,r)$ for any $k$, ...
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### How to proof Lip-constant weakens when size of function class approaches to infinity?

Taken from paper "A Universal Law of Robustness via isoperimetry" by Bubeck and Sellke. Theorem 3. Let $\mathcal{F}$ be a class of functions from $\mathbb{R}^{d} \rightarrow \mathbb{R}$ and ...
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### How to scale a fat matrix to ensure its rows are orthonormal?

I have the following matrix $${\bf B} = \begin{bmatrix} 0&-4.2423&4.2423&1.4871\\ 1.6532&-1.2735& -1.2735&0.0024\\ 0 & -0.2805 & 0.2805 & -0.8823 \end{bmatrix}$$ I ...
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### Let $X$ be a Banach Space.Prove that, $X$ is strictly convex iff every points of $S(X)$ is exposed points of $B(X)$.

Some definitions- $X$ is said to be strictly convex or rotund if for all $x,y\in S(X),\ x\ne y$ we have $\left\lVert\frac{x+y}{2}\right\rVert<1$ A point $x_0\in B(X)$ is said to be a exposed point ...
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### $X$ is strictly convex (rotund) iff for all $x,y\in S(X)$ with $x\ne y$ we have $\lVert x+y\rVert <2$

A Banach Space $X$ is said to be strictly convex or rotund if for all $x,y\in X$ we have $\lVert x+y\rVert<\lVert x\rVert+\lVert y\rVert$ unless $x,y$ are multiple of each other. We have to prove ...
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### How can you proove that every bounded function in $L^1[0;1]$ can be approximated by continuous function in $C[0;1]$?

Here my question, is this true that: Every bounded function in $L^1[0;1]$ can be approximated by continuous functions in $C[0;1]$ It seems to me true as we know that $C[0;1]$ is dense in $L^1[0;1]$, ...
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### Is there a rigorous proof that $L_1$ regularization produces sparse results

I'm looking for a rigorous proof that the $L_1$ regularization really produces sparsity, there are multiple intuitive ones like these which I understand but all of these feel like they just explain ...
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