# Tagged Questions

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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### Expressing continuity in terms of seminorms

Let $X$ and $Y$ be locally-convex topological vector spaces, with topologies given by families of seminorms $(p_i) _{i \in I}$ and $(q_j) _{j \in J}$, respectively. If $L : X \to Y$ is a continuous ...
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### If every real valued continuous function on $A\subseteq \mathbb R^n$ is uniformly continuous , then $A$ is bounded?

Let $A \subseteq \mathbb R^n$ be such that every real valued continuous function on $A$ is uniformly continuous , then $A$ closed and bounded . If $a \in \bar A \setminus A$ , then using the function ...
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### Fixed point problem with matrix

I am looking for guidance for the following fixed point problem. $A$ is an $n\times n$ matrix. The rows of $A=(A_{i})_{i\in N}$ are in the set $A_{i}\in\mathcal{A}$, where $\mathcal{A}$ is the set ...
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### How are absolute value of a field $F$ and norm of vector space $F/F$ related?

So I think it's true that a field $F$ and its respective vector space $F/F$ are isomorphic, since they consist of the same elements, and the operations of $F$ (addition,mult.) and $F/F$ (vector ...
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### Is there only one way to define a norm from an inner product?

Given an inner product $\langle,\rangle$, we can define a norm by $||x|| = \langle x,x \rangle^{\frac{1}{2}}$. My question is, are there other ways to derive a norm from an inner product space and if ...
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### Can someone explain why $\| \cdot \|_p, p = \frac{1}{2}$ isn't a norm?

$\| \cdot \|_p, p = \frac{1}{2}$ or "half norm" is not a norm What is a quick way to verify that it is indeed not a norm?
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### Why in normed vector spaces we can define infinite series but in metric space we can not?

We usually define infinite series by partial sums and an inifinite series is said to converge if its partial sum converges. So, if $X$ is a normed vector spaces and $s_n=x_1+...+x_m$ is a partial sum ...
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### What is a “pseudonorm”?

The following is an excerpt of a note in topological vector spaces. I have tried to search "semi-pseudonorm" on Google but I have got nothing so far. A search with "pseudonorm" returns what we ...
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### Why is the norm ball a square in $\,\mathbb R^2\,$ under $\,l^\infty\,$ norm?

Suppose $\,x = \left(x_1, x_2\right)$, then $\,l^2\,$ norm ball is $\,\left\lbrace x\;\big\vert\;\, \sqrt{\left\lvert x_1 \right\rvert^2 + \left\lvert x_2 \right\rvert^2} \leq 1\right\rbrace$ Easily ...
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### Is the norm ball a set or the boundary of a set?

Recall normed ball in $R^2$ under different norms is typically intuited as follows But looking at someone of the definition of normed ball it seems that it describes a closed set rather than the ...
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### $inf\{||x|| : x \in X , ||F(x)|| = \alpha\} = \frac{\alpha}{||F||}$

If $F \in B(X,Y), F\neq 0$ and $\alpha \geq 0$, then show that $$inf\{||x|| : x \in X , ||F(x)|| = \alpha\} = \frac{\alpha}{||F||}$$ where $B(X,Y)$ is the set of all bounded functions from $X \to Y$ ...
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### Convolution inequality, Gaussian convolution

I'm reading a proof that says for $f\in L^p$ with $p\in[1,\infty)$ we have $\|f\ast p_t-f\|_p\to 0$ as $t\to 0$, where for $t>0$, $p_t(x)=\frac{1}{(2t\pi)^{\frac{d}{2}}} e^{-\frac{\|x\|^2}{2t}}$ is ...
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### Strategy for establishing the triangle inequality of a seminorm

One proof that the $p$-norm $\| x\|_p = (|x_1|^p + \ldots + |x_n|^p)^\frac{1}{p}$ satisfies the triangle inequality exploits the fact that $x \mapsto |x_1|^p + \ldots + |x_n|^p$ is a convex ...
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### How to show continuity of $\cdot$ in a normed vector space?

Let $(V,+,\cdot)$ be a normed vector space with the underlying field $K$ . We have to show that $+,\cdot$ are continuous functions. Since $V$ becomes a metric space under this norm so I can use ...
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### Equivalent norms on $C[0,1]$

For each $f\in C[0,1]$ set $$\|f\|_1 = \left(\int_0^1 |f(x)|^2 dx\right)^{1/2},\quad\quad \|f\|_2 = \left(\int_0^1 (1+x)|f(x)|^2 dx\right)^{1/2}$$ Then prove that $\|\cdot\|_1$ and $\|\cdot\|_2$ are ...
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### Completeness of metric, normed and inner product spaces

For a metric space to be complete, it needs to have all cauchy sequences converge in the metric. 1) For a normed space to be complete, does it need Cauchy sequences to converge in the norm or in the ...
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### Is the spectral radius of a matrix a convex norm of it?

I am wondering if the spectral radius of a matrix is may be some kind of a norm ($l_{\infty}$-norm?) of it and if that is convex. Any pointers to related ideas would be helpful too.
Let $f : \langle V_1, \|\cdot\|_1\rangle \to \langle V_2, \|\cdot\|_2\rangle$ be linear. Then if $f$ is continuous at some $v \in V_1$, then it is continuous on all of $V_1$. Without appealing to ...