# Tagged Questions

Norm is a function on a vector space $X$ which generalizes notion of length of vector in general vector spaces.

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### Is closure of convex subset of $X$ is again a convex subset of $X$?

Today I was going through my text book exercise and got hold of the following question. Let $X$ be a normed linear space with norm $\lVert\cdot\rVert$ and $A$ is a non empty convex subset of $X$ ...
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### Prove that $0 < x < y$ implies $\|x\| < \|y\|$ for any norm.

All vectors are real. Prove that $0 < x < y$ (element-wise) implies $\|x\| < \|y\|$ for any norm. This is probably very basic, but I don't seem to get the hang of it. Edit: it turns out this ...
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### The difference between $L_1$ and $L_2$ norm?

I have been trying to understand what is the difference between $L_1$ and $L_2$ norm and cant figure it out. In this webpage I got a clear understanding of why we would use $L_1$ norm (scroll down ...
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### Norm of the integral operator in $L^2(\mathbb{R})$.

Suppose we have an integral operator $A$ such that $$Af(x) = \frac{1}{\sqrt{2\pi}}\int\limits_{\mathbb{R}}e^{-\frac{(x-y)^2}{2}}f(y)dy$$ To find $\|A\|$ we can use the unitary Fourier transform $F$, ...
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### Does $\|A\|_2\leq \|B\|_2$ imply $\|CA\|_2\leq \|CB\|_2$ or $\|AC\|_2\leq \|BC\|_2$?

Assume matrices $A$, $B$, and $C$ are of same dimensions, does $\|A\|_2\leq \|B\|_2$ imply $\|CA\|_2\leq \|CB\|_2$ or $\|AC\|_2\leq \|BC\|_2$? $\|A\|_2$ denotes by $\lambda_{max}\sqrt{A^TA}$, and ...
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### Proof that the Euclidean norm is indeed a norm

I apologize beforehand for this question. Its embarrassing I know. Anyway, here we go. Recall: $$\| x \|^2_{\mathbb{R}^2} = \sum^{n}_{i = 1} x^2_{i}$$ How do we prove its a norm? Well if its a ...
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### Is it true that $2^p\|f\|^p_{L_p}+2^p\|f'\|^p_{L_p} >2\|f\|^p_{L_\infty}$?

For any $C^1$ function defined in $(0,1)$, is it true that $$2^p\|f\|^p_{L_p}+2^p\|f'\|^p_{L_p} >2\|f\|^p_{L_\infty}$$ If it is true, how to prove it?
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### Operator Norm of a Matrix composed of Standard Basis and Fourier Basis

Let $\mathbf{A}_n$ be an $n\times 2n$ matrix (where $n=2^k$) composed of Fourier basis and standard basis; that is, $$\mathbf{A}_n = \begin{bmatrix}\mathbf{I}_n & \mathbf{F}_n\end{bmatrix}$$ ...
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### why is $\ell_0$ a pseudo-norm?

Let $\mathbf{x}$ be a vector in $\mathbb{R}^n$. We define the $\ell_0$ pseudo-norm by:$$\|\mathbf{x}\|_0=\#\left\{i : \mathbf{x}_i\neq0\right\}$$ Why $\|\cdot\|_0$ is not properly a norm?
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