# 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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### Approximate $L^2$ norm of sum of vectors

Suppose I need to compute the $L^2$ norm $| \cdot |$ of the sum of $n$ vectors $v_i \in \mathbb{R}^d$. Denote $k = | \sum_{i=1}^n v_i |$, which can be achieved via adding all the vectors and then ...
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### Finite Dimensional Subspace of a Normed Linear Space is complete.

I am working through the wiki proof that every finite dimensional subspace of a normed linear space is closed. The proof goes as follows: Let $(V ,\| \cdot \|)$ be a normed linear space. Let $W$ be a ...
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### 2-norm of transpose proof

I don't understand the proof of ‖x‖2=‖xT‖2. I know you can use the Cauchy-Schwartz inequality and that you can prove it by first showing ‖xT‖2 ⩽ ‖x‖2 and then ‖x‖2 ⩽ ‖xT‖2. I get proving both ...
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### Expected value of a euclidean 2 space norm [closed]

Say we have $A=(a_1,a_2)$ and $B=(b_1,b_2)$ which are independent and uniform on say $[0,x]^2$ how would we find $\mathbb{E}(||A-B||^2)= \mathbb{E}((a_1-b_1)^2+(a_2-b_2)^2)$.
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### Show the continuity of a function between vector space

I consider the function defined by $\phi(f)$ which associates to each function $f$ from $C([0,1],\mathbb{R})$ the function $\phi(f)(x) = \int_{0}^{x} tf(t)dt$ from $[0,1]$ to $\mathbb{R}$. My idea is ...
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### Proof by contradiction that $(\ell^p)^* \subseteq \ell^q$

For $p \in (1, \infty)$, let $\ell^p = \{(x_n) : \sum |x_n|^p < \infty\}$ and let $(\ell^p)^*$ denote the dual space of bounded linear forms on $\ell^p$ $(\mathbb{F} = \mathbb{R})$. It is ...
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