# Tagged Questions

1answer
40 views

### Finite-dimensional subspace normed vector space is closed

I know that probably this question has already been answered, but I'd like to present my attempt of solution. Let $(E,\|\cdot\|)$ be a normed vector space and let $F\subseteq E$ be a ...
1answer
43 views

### Surjection of norms

Let $V$ be an infinite dimensional $\mathbb{C}$ (or $\mathbb{R}$) vector space. Suppose there exists two norms on $V$ such that \begin{equation*} \| \cdot\|_1 \leq \| \cdot \|_2. \end{equation*} Is ...
3answers
57 views

### Let $V$be a vector space. Prove/Disprove: There is a norm $\|\cdot\|$, such that all subsets of $V$ are open sets in $(V,\|\cdot\|)$.

The Assignment: Let $V$ be a vector space over $\mathbb{R}$ with $V \not= \{0\}$. Prove or disprove: There is a norm $\|\cdot\|_d$ on $V$, such that all subsets of $V$ are open sets in ...
0answers
33 views

### Convergent sequences in normed vector spaces

Consider a normed vector space $X$. I have a couple of questions regarding convergent sequences and subspaces of $X$(topological subspaces, not necessarily linear subspaces). Let $W$ be a topological ...
0answers
47 views

### Complete Normed Space => Uncountable Hamel basis not by Baire

I need to show that a complete normed space X has no countable Hamel basis. One possibility is to with Baire's theorem. I, however, try to give an explicit sequence, namely: For a contradition, let ...
0answers
21 views

2answers
82 views

### Proving the normed linear space, $V, ||a-b||$ is a metric space (Symmetry)

The following theorem is given in Metric Spaces by O'Searcoid Theorem: Suppose $V$ is a normed linear space. Then the function $d$ defined on $V \times V$ by $(a,b) \to ||a-b||$ is a metric on $V$ ...
1answer
87 views

### Is a normed topological space metrizable?

As stated in the title: If there is a norm on a topological space, then we get a metric induced by the norm. Is this true?
1answer
84 views

### Convergence of coordinates to zero

Consider a normed finite-dimensional vector space $V$ with some norm $|| .||$ Say a sequence of vectors in this vector space $v_m \rightarrow 0$ where $0$ is the zero vector. Let ...
2answers
129 views

### Picard iterations of a matrix

I need help with this problem. I think i got the first three questions of the exercise, but i'm stuck at the fourth one. We consider the map $T:{\mathbb{R}^2}\longrightarrow{\mathbb{R}^2}$ defined ...
2answers
67 views

### Linear functional $\mathscr{L}(E,F)$

Let $\mathscr{L}(E,F)$ denote the space of all linear functionals from $E \to F$. Let $\mathscr{C}(E,F)$ denote the space of continuous linear functionals from $E \to F$. My question: How to prove ...
1answer
90 views

1answer
317 views

### A stronger statement of Riesz's lemma

Riesz's lemma state that If $Y$ is a proper, closed subspace of a normed space $X$, then for any $\epsilon>0$, there exists $x$ in the closed unit ball of $X$ such that $d(x,Y)>1-\epsilon$. ...
5answers
184 views

### Is closure of convex subset of $X$ is again a convex subset of $X$?

Today I was going through my text book exercise and got hold of the following question. Let $X$ be a normed linear space with norm $\lVert\cdot\rVert$ and $A$ is a non empty convex subset of $X$ ...
2answers
867 views

### Is closure of linear subspace of X is again a linear subspace of X??

Let $X$ be a normed linear space with norm $||\cdot||$ and $A \neq \emptyset$ is a linear subspace of $X$. Prove that $\bar{A}$ is also a linear subspace of $X$.
1answer
38 views

### Replacing one of the conditions of a norm

Consider the definition of a norm on a real vector space X. I want to show that replacing the condition $\|x\| = 0 \Leftrightarrow x = 0\quad$ with $\quad\|x\| = 0 \Rightarrow x = 0$ does not alter ...
2answers
205 views

### If $V \times W$ with the product norm is complete, must $V$ and $W$ be complete?

Let $V,W$ be two normed vector spaces (over a field $K$). Then their product $V \times W$ with the norm $\|(x,y)\| = \|x\|_V + \|y\|_W$ is a normed space. Using this norm it's easy to show that if ...
1answer
112 views

### Practical implications of a vector space being a topological vector space

I have a space $V$ and I lately discovered that it's a topological vector space. What are the practical implications of that?
3answers
1k views

### Interior of a Subspace

There is a conjecture: "The only subspace of a normed vector space $V$ that has a non-empty interior, is $V$ itself." (here, the topology is the obvious set of all open sets generated by the metric ...
2answers
359 views

### Norm-preserving map is linear

How can one show that a norm-preserving map $T: X \rightarrow X'$ where $X,X'$ are vector spaces and $T(0) = 0$ is linear? Thanks in advance.
2answers
4k views

### Difference between metric and norm made concrete: The case of Euclid

This is a follow-up question on this one. The answers to my questions made things a lot clearer to me (Thank you for that!), yet there is some point that still bothers me. This time I am making ...