# 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.

72 views

34 views

### Is the result true when the valuation is trivial and $\dim(X)=n$?

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 ...
31 views

### Does a convergent sequence of norms of a vector space always converge to a norm?

I thought that just as the sequence of norms $||x||_p :\mathbb{R}^n \mapsto \mathbb{R}$ converges to $||x||_{\infty}$ maybe there is some result that proves that every convergent sequence of norms of ...
19 views

### Computing the norm of $\varphi ((x_n)_{n \in \mathbb{N}})=\sum_{n=1}^{\infty} a_n x_n$

Let $p,q>1$, $\frac{1}{p}+\frac{1}{q} = 1$ and $(a_n)_{n \in \mathbb{N}}\in l_q$. Show that $\varphi : l_p \rightarrow \mathbb{R}$, $\varphi ((x_n)_{n \in \mathbb{N}})=\sum_{n=1}^{\infty} a_n x_n$ ...
35 views

Is there a general rule to simplify things like $<x,y> - <x,z>$ or generally $<.,.> \pm <.,.>$ I cant find anything in my notes that talks about this.
23 views

### Normed spaces: Sum of closures is a subset of the closure of the sum

Let $E$ be normed space and $A,B\subset E$. Show that $\overline{A}+\overline{B}\subset \overline{A+B},$ where $A+B=\{ a+b :a\in A \text{ and } b\in B \}$. Now I know this is true if $A$ and $B$ ...
78 views

### Norm of linear combination of vectors in the “same general direction”

Let $X$ be a normed vector space (a Banach space if necessary) and $x, y \in X$ such that $||x|| \leq ||x + y||$ and $||y|| \leq ||x + y||$. (Intuitively, I take this to mean $x$ and $y$ are in the "...
36 views

### 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 $\lim_{n\to\infty} f_n(t) = f(t)$. (...