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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Intuition of a norm vector space/ infinite dimensional vector space

I'm finding it terribly difficult to build an intuition of what a norm vector space and an infinite dimensional vector space is. There aren't any good notes online that builds the intuition-most ...
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34 views

Matrix tensor indices

Suppose I have an orthonormal basis $$B = \left \{ u_{i} \right \}_{i=1}^{\infty}$$ Then for a matrix $K$, do I represent it as $$K = \sum_{j,k=0}^{\infty}k_{jk}\left ( u_{i}\bigotimes u_{j} \...
2
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1answer
39 views

$f(x)$ be the characteristic polynomial of a matrix $A \in M_n(\mathbb R)$ ; then is it true that $f(1)=1+\operatorname{trace}(A)+O(\|A\|^2)$?

Let $f(x)$ be the characteristic polynomial of a matrix $A \in M_n(\mathbb R)$; then is it true that $f(1)=1+\operatorname{trace}(A)+O(\|A\|^2)$ ? I need a proof if it is true ; or any modification ...
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47 views

Problem in showing that a norm is a norm on one space, but not on another.

I have the following question from a past paper: "Consider the two sets, $$A=\{g\in C^1([0,1]):g(0)=g(1)=0\}$$ and, $$B=\{g\in C^1([0,1]):g'(0)=g'(1)=0\}$$ both subsets of the vector space $C^1([...
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75 views

Find a norm so that its closed unit ball is the area between $y=x^2-1$ and $y=1-x^2$

As the title specifies, I need to find an explicit formula for a norm $|||\cdot|||$ so that: $$B_{|||\cdot|||}=\{\mathbf{x} : ||| \mathbf{x}|||\le1 \}$$ where $\mathbf{x}=(x,y)\in\mathbb R^2$, is ...
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32 views

Why is convexity of $S$ needed in these questions?? (Bachman &Narici, Q-17,18)

Let $X$ be a Hilbert Space and let $\{S\}$ be a Convex set in $X$. Let $d=\inf_{x \in S}\|x\|$ . Prove that, if $\{x_n\}$ is a sequence of elements in $S$ such that $\lim_n \|x_n\|=d$, then $\{x_n\}$ ...
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2answers
90 views

Find the precise conditions under which we have $\|x+y\| = \|x\|+\|y\|$

In $\mathbb{R}^n$ consider (the norm infinity) $\|x\|=\max|x(i)|$ where $1\leq i\leq n$. Find the precise conditions under which we have $\|x+y\|=\|x\|+\|y\|$. Thank you for your helping. :)
2
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2answers
45 views

The norm of the extension of an operator

If $T_0$ is the identity operator from $E$ to $E$, where $E$ is a subspace of $F$ and both of them are Banach spaces (maybe not needed). If the bounded linear operator $T$ is an extension of $T_0$ ...
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1answer
168 views

A proper subspace of a normed vector space has empty interior.

In a vector normed space $E$, prove that all vectorial subspace $F\neq E$ has a interior empty. My approach:We consider, the open ball $B\subset F$, with $F$ proper subspace of $E$. If $x\notin F$, ...
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3answers
40 views

Completness of Normed spaces.

I want to prove the following proposition If $(X,||\cdot||)$ and $(X,||\cdot||')$ are homeomorphic, then $(X,||\cdot||)$ is complete if and only if $(X,||\cdot||')$ is complete. So, I only know the ...
2
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1answer
139 views

$\ell^p\!,$ for $p\neq2$, is not an inner product space. [duplicate]

Consider sequence spaces of the form, $$\ell^p=\Big\{x=\left(x_j\right)_{j=1}^\infty \mathrel{}\big|\mathrel{} \sum_{j=1}^\infty \left\lvert x_j \right\rvert ^p\lt\infty\Big\}$$ for $1\le p\lt\...
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1answer
64 views

Point about the theorem and proof of the inner product being a continuous function.

In moving to show that the inner product, $\langle\cdot,\cdot\rangle$ is a continuous function I have the following theorem in my notes (also on page 59 of "Linear Functional Analysis", Rynne and ...
2
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1answer
29 views

Can absolute scalability be 'relaxed' to an equivalent condition in the properties of a norm?

All norms on a vector space $V$ must satisfy for any $x\in V$ $$\Vert \alpha x \Vert = \vert \alpha \vert \Vert x \Vert $$ for any scalar $\alpha\in R$. However, I've been told that an equivalent ...
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0answers
62 views

A consequence of “every norm is equivalent to the sup-norm” in a finite dimensional normed vector space

So, I am reading the following proposition in Neukirch's Algebraic Number Theory: Proposition. Let $K$ be a complete field with respect to the valuation $|\:\: |$ and let V be an $n$-dimensional ...
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45 views

Questions about the finite-dimensional normed space of polynomials of degree at most two.

Take $X:=P_2([0,1])$, the polynomials of degree at most $2$ over $[0,1]$ and consider the $2$-norm on this space. For any $x\in X$ we have that, $$\|x\|_2=\left(\sum_{i=1}^n|x_i|^2\right)^{1/2}$$ ...
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1answer
32 views

X and Y are normed linear spaces over the same field $F(=\mathbb{C}/\mathbb{R})$, both having the same finite dimension $n$.

X and Y are normed linear spaces over the same field $F(=\mathbb{C}/\mathbb{R})$, both having the same finite dimension $n$. I need to show that $X$ and $Y$ are topologically isomorphic ( A ...
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1answer
55 views

How does sketching norms show that they are equivalent?

I have the following statement in my notes: "You might want to check by drawing the sets of all $x\in\mathbb R^2$ such that $\|x\|_1=1$,$\|x\|_2=1$,$\|x\|_\infty=1$ that indeed these norms are ...
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34 views

Can We Always Realize the Value of the Quotient Norm. [duplicate]

Let $(V, \|\cdot\|)$ be a Banach space over $\mathbf R$ and $W$ be a closed subspace of $V$. We know that $V/W$ becomes a normed linear space under the quotient norm $\|\cdot\|_q$ defined as $\|v+W\|...
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1answer
55 views

Understanding convergence in normed spaces and the language used when talking about norms.

We have the following definition about convergence in a normed space: "Let $(x_n)_{n=1}^\infty$ be a sequence in a normed space $(X,\|\cdot\|)$. We say that $x_n\to x$ in $X$ if, $$d(x_n,x)\equiv \|...
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521 views

Examples of infinite dimensional normed vector spaces

In my notes on functional analysis it mentions that $C([0,1]),\ell^p$ and, $\ell^\infty$ are normed vector spaces, and gives some examples of norms that we can define on them. However, it then simply ...
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26 views

Need help with this proof, theory of finite change.

Theory: If $f : [a,b] \to X $ is differentiable on (a,b) and continuous on [a,b] in $X$, a normed vector space upon $ \langle , \rangle$ then: $$|f(b)-f(a)| \leq \sup_{a <c<b}{\|f'(c)\|}(b-a)$$ ...
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1answer
24 views

$| \langle a , i \rangle| \leq \| a\|$ if $\|i\|=1$ this space is a normed vector space upon $\langle , \rangle$ . Why does this apply?

I tried over Cauchy Schwarz to conclude, but could not. Anyone see why this is ? The term: normed vector space upon $\langle , \rangle$ i hear for the first time, Im assuming it means that: $$\|a \| = ...
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91 views

Equivalence of Norms and Open Mapping Theorem

Let $V$ be a vector space with two norms $||\quad||_{1}$, $||\quad||_{2}$, making $V$ a complete normed vector space. Assume $\exists C$ (constant) such that: $||v||_{2} \leq C||v||_{1}, \forall v \...
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1answer
46 views

Operator norm and equivalent definitions

From the definition of the operator norm, we have: $||T||_{op}=\inf\{c\in \mathbb{R}^+:||Tv||\leq c||v||, v \in V\}$ If $T: V \rightarrow W$ is a linear map between two normed vector spaces. I have ...
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20 views

A metric space and polytopes

suppose we define a metric space $L_2(p)$ induced by a duality pair given by $\langle x,y\rangle =\sum_{j,k} p(j,k)[ x_1^j y_1^j+ x_2^k y_2^k]$ Where $x=(x_1,x_2)$ and $y=(y_1,y_2)$ and $p$ is a ...
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31 views

$f'(x;y)=0$ for every $x$ in an open convex set and for every vector $y$ ; then to show $f$ is constant on $S$

Let $f:\mathbb R^n \to \mathbb R$ be a map , $S$ be an open convex set in $\mathbb R^n$ such that for every $x \in S$ and $y \in \mathbb R^n$ , $f'(x;y)$ exists and equals $0$ ; then how to show that $...
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2answers
119 views

What is the implication that $\| \cdot \|_2$ and $\| \cdot \|_\infty$ are equivalent norms on $\mathbb{R^2}$

Given $\mathbb{X}$ = $\mathbb{R^2}$, consider $\| \cdot \|_2$ and $\| \cdot \|_\infty$ We can show that $\| x \|_\infty \leq \| x \|_2 \leq \sqrt2 \| x \|_\infty$ Hence $\| \cdot \|_2$ and $\| \...
6
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1answer
93 views

Norm $\Vert \cdot \Vert$ on the symmetric group $S_n$

If we define a real valued function $\Vert \cdot \Vert$ on the $n^{th}$ order symmetric group $S_n$ satisfying following conditions $$\begin{align} & \|x\|=0\iff x=\omega\,\,\,(\text{identity ...
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1answer
46 views

Is the closure of every bounded convex set , with non-empty interior , in $\mathbb R^n (n>1)$ homeomorphic to a closed ball?

Is the closure of every bounded convex set , with non-empty interior , in $\mathbb R^n (n>1)$ homeomorphic to a closed ball (by closed ball I mean $B[a,r]:=\{x \in \mathbb R^n : d(x,a)\le r\}$ , ...
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1answer
31 views

Which subsets of a given real linear space $V$ are open unit balls with respect to some norm on the space?

Which subsets of a given real linear space $V$ are open unit balls with respect to some norm on the space? My textbook (Metric Spaces - Michael Searcoid) gives the following hints to answer the above ...
2
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1answer
65 views

Specific question on $l^p$ spaces and its dual in weak * topology

I am covering now Lp spaces in my summer real analysis course and this problem from Folland related to the dual of Lp stumped me hard, it is problem 19 chapter 6 reads as follows: We define $ \...
4
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2answers
78 views

Parallelogram law in $L_1$ space

Exercise 5.5 from Capinski's and Kopp's book "Measure, Integral and Probability" asks to show that it is impossible to define an inner product on the space $L^1([0,1])$. In order to get this result we ...
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0answers
32 views

Generate complete functions set in hilbert space

It is just a curiosity, but is there a general method (or a class of methods) that allows to derive orthonormal complete function set for a given hilbert space? (Except Gram Shmidt algorithm and ...
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1answer
39 views

How are $M_n(A)$ and $M_n(\mathbb{C})\otimes A$ identified as Banach algebras?

When $A$ is a Banach algebra, one can make $M_n(A)$ into a Banach algebra by equipping it with various norms. One can also make the algebraic tensor product $M_n(\mathbb{C})\otimes A$ into a Banach ...
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1answer
49 views

Is $C[0,1]$ equipped with $\lVert \cdot \rVert_1$ a countable union of nowhere dense sets?

Let's consider the space of all continuous function $C[0,1]$ on the intervall $[0,1]$. But instead of using the usual supremum norm we use the $L^1$-Norm $: \lVert f \rVert_1=\int_0^1 \lvert f(x) \...
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1answer
70 views

The spectrum of a polynomial of an operator, question about proof, why are the operators invertible?

I have a question about a proof. In the proof $\sigma(T)$ is $\{\lambda \in\mathbb{C}: T-\lambda I\text{ is not invertible}\}$. In the proof they use this lemma: Here is the proof, my problem is ...
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18 views

$A$ be a convex subset and $D$ be dense in a real NLS ; then for every non-empty open subset $U$ of the NLS , $U\cap D\cap V \ne \phi$?

Let $A$ be a convex subset of a real normed linear space $V$ and $D$ be a dense subset of $V$ ; then is it true that for every non-empty open subset $U$ of $V$ , $U\cap D\cap A $ is non-empty ?
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37 views

Complement of the union of countably many , mutually disjoint , non-empty open balls in $\mathbb R^n , (n >1) $ is path connected?

Let $n \ge 2$ and $\{B_m\}_{m=1}^\infty$ be countably infinitely many , mutually disjoint , non-empty open balls in $\mathbb R^n$ , then is $\mathbb R^n \setminus \cup_{m=1}^\infty B_m$ path-...
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64 views

Prob. 10, Sec. 4.5 in Kreyszig's Functional Analysis: How to relate this result to solution of equations?

Let $T \colon X \to Y$ be a bounded linear operator, where $X$ and $Y$ are normed spaces, both real or both complex; let $B$ be a subset of the dual space $X^\prime$ (i.e. the normed space of all the ...
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2answers
54 views

Prove that the linear transformations are the same.

I have this lemma: If X is a complex inner product space and $S,T \in B(X)$ are such that $(Sz,z)=(Tz,z)\forall z \in X$, then $S=T$. $B(x)$ is the set of bounded linear operators from X to X. ...
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1answer
53 views

Prob. 3, Sec. 3.2 in Kreyszig's Functional Analysis Book: Is the space of all polynomials of a fixed degree complete? [duplicate]

Let $n$ be a given natural number, and let $X$ denote the vector space consisting of the zero polynomial and of all polynomials of degree at most $n$, with real or complex numbers as co-efficients, ...
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1answer
60 views

Image of a function with small BMO norm

This is a question related with the regularity of harmonic maps. Let $N\geq 1$ and $f:\mathbb{R}^N\to \mathbb{S}^2$, where $\mathbb{S}^2=\{x\in \mathbb{R}^3 : \|x\|=1\}$. Assume that the BMO semi-...
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2answers
54 views

Show that $\| x-x_j \|_2^2$ is equal to

I have a simple question. $$\|x-c_j\|_2^2 = x^T x-2x^T c_j+c_j^T c_j$$ Where $x,c_j\in\mathbb{R}^n$, $j=1,2,\ldots,n$. I know that $$\|x-c_j\|_2^2=(x-c_1)^2+(x-c_2)^2+\cdots+(x-c_n)^2.$$ But I don't ...
3
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0answers
80 views

When weakly compactness implies compactness?

Let $A$ be a Banach space. The weak topology on $A$ is a topology which produced by the following family of seminorms: $~~~~~~~~~~~~~~~~~~~~P_f(x)=|f(x)|,\qquad$ where $f\in A^*$ and $A^*$ is dual ...
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1answer
39 views

How to show a norm identity of a weighted sum

I ran across the following identity while reading up on norms. It deals with the square of the $2$-norm of a convex combination. That is, for all $x,y,\in\mathbb{R}^{n}$ and $\rho \in [0,1]$: \begin{...
0
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1answer
48 views

Dual norm of $L_p$ space

Given $R^n$ is equipped with the norm $||x|| = (\sum_{k=1}^{n} |x_k|^p)^{\frac{1}{p}}$ for some $p ≥ 1$, what is the induced norm on the conjugate (dual) space? I couldn't figure out how to prove ...
3
votes
1answer
86 views

How is this the Open Mapping Theorem?

My book has this theorem which it has stated as the Open Mapping Theorem: Suppose X and Y are Banach spaces and $T \in B(X,Y)$ is surjective. Let: $L=\{T(x): x \in X \text{ and } \|x\|\le 1\}...
2
votes
1answer
52 views

Are invertible linear operators of bounded linear operators also bounded?

I have this definition in my book: Definition: Let X,Y be normed linear spaces. An operator $T \in B(X,Y)$ is said to be invertible if there exists $S \in B(Y,X)$ such that $ST=I_X, TS=I_Y$, ...
3
votes
2answers
101 views

Is $L^1(X) \cap L^2(X)$ a closed subspace of $L^2(X)$ and $L^1(X)$?

Suppose that $X$ be a locally compact Hausdorff space. Could we say that $L^1(X)\cap L^2(X)$ is closed subspace of $L^1(X)$ and $L^2(X)$?
2
votes
1answer
77 views

Prove continuity for a given norm

I struggle with this exercise from an analysis 2 book I use for self study: Let V := $C^1([0,1]; \mathbb{C})$ the vector space of continously differentiable functions from $[0,1]$ to $\mathbb{C}$ ...