# 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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### Representation of the elements of a normed vector space that has a dense subset.

Suppose that $U$ is a normed vector space and $S\subset U$ is dense.prove that every element of $U$ can be written as an absolutely convergent series of the finite linear combination of the ...
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### Prove a sequence in $\ell^\infty$ is bounded but not Cauchy

$e_n$ in $\ell^\infty$ is the sequence whose $n$th entry is 1, and all others are 0. Show that ${e_n}$ from $n=1$ to infinity is bounded but not Cauchy. I'm really not certain of the whole concept of ...
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### How do completeness and being closed differ in a subspace?

I am meant to prove that if $E$ is a finite dimensional subspace of a normed space $X$, then $E$ is a closed subspace. Now I know that if $E$ is a finite dimensional subspace of $X$, then $E$ is ...
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### Vector space containing vectors of infinite norm not complete?

Let $V$ be a vector space such that there is a $v \in V$ with $\|v\|_V = \infty$. Can you conclude from this that $V$ is not complete, i.e. that there is a Cauchy sequence in $V$ which does not ...
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### Proving that a subspace of a normed vector space is closed

Question: Let X be a normed vector space. If M is a closed subspace of X and x ∈ X − M then M + ℂx is closed where M + ℂx = { y + λ x : y ∈ M , λ ∈ ℂ } There's a theorem from Folland's Real Analysis ...
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### Closed set in normed vector space

Is it true that a subspace M of a normed vector space X is closed if the limit of every sequence in M is contained in M? Whether or not X is complete? Are there alternative characterizations of ...
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### Does $\|\sum_{i=1}^{\infty}x_i\| \leq \sum_{i=1}^{\infty}\|x_i\|$ hold for a norm?

In a normed linear space $(X,\|\cdot\|)$, by definition we have $\forall x,y \in X$, $$\|x+y\| \leq \|x\|+\|y\|.$$ My question is, is it true (or does the definition of norm imply) that for a sequence ...
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### Normed Quotient Space

If $X$ is a normed vector space and $M$ is a proper closed subspace, I want to show that for any $\epsilon>0$ there exists an $x\in X$ such that $\|x\|=1$ and $\|x+M\|\geq 1-\epsilon$. Is there ...
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### Normed Vector Space & Compactness

Let $V$ be a real normed vector space. Let $K$ be a compact set from $V$. Show that the set $2K =$ {2$x$: $x \in K$} is also compact. A topological space is compact if every open covering has a ...
Let $V$ be a real normed vector space. Suppose that $A$ is an open set from $V$. Show that the set $\frac12 A = \left\{ \frac12 x \, : \, x \in A \right\}$ is also open. Let $V$ be a complex vector ...