# 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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### lim r(t) = L implies lim ||r(t)|| = ||L||

The question is: if $\lim_{t\rightarrow t_0}\vec r(t) = \vec L$ show that $\lim_{t\rightarrow t_0} \|\vec r(t)\| = \|\vec L\|$ So here's where I am so far: let $\vec r(t) = (f_1, f_2,...f_n)$ be ...
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### Solution of Volterra equation in $L_\infty$?

I'm having some trouble proving the following (if it is at all true?). Consider a time-varying Volterra equation $$F(x, \xi) = f(x, \xi) + \int_\xi^x G(x, s) F(s, \xi) ds$$ for some (known) ...
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### Is the result true when the valuation is trivial?

Here I proved the following result: Proposition: Let $(K,|\cdot|)$ be a valued field with non-trivial valuation and let $X$ be a vector space over $(K,|\cdot|)$. Two norms $p_1,p_2$ on $X$ are ...
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### Distance Between Vectors

Generally, my main question is how to compute distance between two vectors. I'm aware of the formula $d=\| v-u\|$ where $v,u$ are two vectors, and $d$ donates the distance between them. More ...
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### Prove that $(V,\lVert\cdot\rVert_\infty)$ is a Banach space

$X=C[0,1]$ (i) Prove that if the sequence $(f_n)_{n\ge1} \subseteq X$ converges to $f \in X$ in the supremum norm, then for each $t\in[0,1]$ one necessarily has $\lim_{n\to\infty} f_n(t) = f(t)$. (...
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### regularized least squares (L1 norm)

My objective function that is to be minimized is as follows: $$\|y-Ax\|_2^2 + \alpha\|Lx\|_1$$ where $L$ is the gradient operator. Now this problem seems convex because the first term is quadratic ...
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### Derivative of squared norm of a complex function

Assuming $f(x)$ is a complex function, its squared norm is defined as $$|f(x)|^2 = f(x) . f^*(x)$$ What is derivate of $|f(x)| ^ 2$?
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### Hamacher product in fuzzy logic

I was reading the article here Hamacher product is In page 41 in the article it describes three properties of Hamacher product. I do not understand the difference in notations and meaning of the ...
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### Spectral Norm Proof

I don't really understand the question below: A matrix can have small eigenvalues but large spectral norm (i.e., largest singular value). Find such a matrix A of proper dimension so that all of its ...
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### What does this (double absolute value like) notation mean?

Here, $$\left\lVert\frac{\partial\bf x}{\partial s}\times\frac{\partial\bf x}{\partial t}\right\rVert$$ the inside will at last be a vector. and two absolute value signs have covered it. what does ...
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### Check $F_2(f)=f'(1)+f(1)$ is a linear functional

Let $(X,\|\cdot \|)=(C[0,1],\|\cdot \|_{\infty})$ and $F_1 :X \rightarrow \mathbb R$ be defined by $$F_1 (f)=\int _{1/2}^{3/4} f(t) dt$$ $F_1$ is a continuous linear functional. Lets consider the ...
### Show $L$ is a closed linear subspace of $H$
Let $H$ be a Hilbert space and linear operator $P:H \rightarrow H$ satisfy $<Px,y>=<x,Py>$ for all $x,y \in H$ and $P^2=P$. We can use the fact that $Px \perp (x-Px)$ for every $x \in H$ ...