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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2
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0answers
282 views

Dual norm of the matrix $L^1$ norm is infinity norm (and vice versa)

Recall that for a given norm $\|\cdot\|$ on $\mathbb{R}^n$, the dual norm is defined as a function $\|\cdot\|_*: \mathbb{R}^n \rightarrow \mathbb{R}$ with: $\|y\|_* = \max \limits_x \{x^Ty: \|x\|\le1 ...
2
votes
1answer
34 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
votes
2answers
35 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 ...
0
votes
1answer
112 views

Trouble in proving that $\|x\|_p = \max|x_j|$

We define p-norm in this way: $\|x\|_p = \{\sum ^N_j=_1|x_j|^p\}^ {1\over p}$ We know that It change to $\|x\|_p = \max|x_j| $ when $ p \to \infty $ How can I prove this ?
1
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0answers
21 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 ...
2
votes
0answers
11 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 ...
7
votes
1answer
86 views

How does $\lim A_n$ being not invertible imply $\sup_n\|A_n^{-1}\|=\infty$?

Consider a sequence of operators $\{A_n\}_{n=1}^{\infty}\subset B(X,Y)$, where $X,Y$ are normed vector spaces and $B(X,Y)$ denotes the space of bounded linear operators from $X$ to $Y$. Assume that ...
1
vote
1answer
32 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) ...
3
votes
1answer
32 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 $\mathbb R^n \setminus \cup_{m=1}^\infty B_m$ ...
0
votes
1answer
23 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 ...
-3
votes
2answers
26 views

$A$ is a convex subset with non-empty interior and $D$ is dense in $\mathbb R^n$ ; then $\mathbb R^n$ , $U\cap D \cap A \ne \phi$? [on hold]

Let $A$ be a convex subset , with non-empty interior , of $\mathbb R^n$ and $D$ be a dense subset of $\mathbb R^n$ ; then is it true that for every open subset $U$ of $\mathbb R^n$ , $U\cap D \cap A$ ...
1
vote
1answer
39 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 ...
0
votes
3answers
12 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 ?
0
votes
0answers
28 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 ...
3
votes
2answers
45 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. ...
1
vote
1answer
28 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, ...
12
votes
3answers
1k views

Does every $\mathbb{R},\mathbb{C}$ vector space have a norm?

Is there a canonical way to define on any vector space over $\mathbb{K}=\mathbb{R},\mathbb{C}$ a norm ? (Or, if there isn't, can someone give me an example of a vector space over $\mathbb{K}$ that is ...
1
vote
2answers
48 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
votes
0answers
63 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 ...
0
votes
1answer
25 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 $\rho \in [0,1]$: ...
0
votes
0answers
7 views

Norm on $M_n((M_k(A))^+)$ where $A$ is a Banach algebra

If $A$ is a Banach algebra (either unital or non-unital), can we norm $M_n((M_k(A))^+)$ by regarding matrices in there as matrices in $M_{nk}(A^+)$ and taking one of the commonly used norms on the ...
2
votes
1answer
30 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 $ST=I_X, TS=I_Y$, ...
0
votes
1answer
17 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
37 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 ...
2
votes
1answer
280 views

Space of finite dimensional subspaces is separable

In Bernard Maurey's paper "A Note on Gowers' Dichotomy Theorem" at the top of the 7th page, the following fact is stated that I'm not able to prove: Let $X$ an infinite dimensional separable normed ...
3
votes
2answers
92 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
22 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 $\mathbb{C}$ ...
0
votes
1answer
30 views

How can I solve the following exercise

Prove that a linear operator $T:X\rightarrow Y$ is bounded if and only if it maps sequences that converge to zero to bounded sequences .
0
votes
1answer
39 views

Prob. 9, Sec. 4.3 in Kreyszig's Functional Analysis Book: Proof of the Hahn Banach Theorem without Zorn's Lemma

Here's Theorem 4.3-2 (i.e. the Hahn Banach theorem for normed spaces): Let $f$ be a bounded linear functional defined on a subspace $Z$ of a normed space $X$. Then there exists a bounbed linear ...
1
vote
0answers
24 views

Prob. 6, Sec. 4.3 in Kreyszig's Functional Analysis Book: What are all possible extensions?

Here's Theorem 4.3-2 in Introductory Functional Analysis With Applications by Erwine Kreyszig: Let $X$ be a normed space, let $Z$ be a subspace of $X$, and let $f$ be a bounded linear functional ...
0
votes
1answer
26 views

Norm of Matrix vector product

Given a vector $x \in \mathbb{R}^n$ we know the following inequality holds for the product of the vector $x$ and a matrix $A \in \mathbb{R}^{m\times n}$ i.e., $Ax=y$ where $y \in \mathbb{R}^m$ ...
0
votes
1answer
41 views

Cauchy Schwarz inequality using L1 norm

1) Cauchy-Schwarz inequality states that the absolute value of vector inner product is always less or equal to product of norms of individual vectors i.e., $|a^Tb|\leq\Vert a\Vert_2 \Vert b\Vert_2$. ...
1
vote
3answers
46 views

A nice way to see that $(\ell^\infty,d_\infty)$ is not separable

Let $(X,d)$ a metric space such that $\exists A\subseteq X$ uncountable and $\exists \epsilon\gt0$ such that $\forall x,y \in A,x\neq y\Rightarrow d(x,y)\gt\epsilon$. Prove that $X$ is not a ...
0
votes
0answers
39 views

What is the relation between the matrix of a bounded linear operator and that of its adjoint?

Let $X$ and $Y$ be finite-dimensional normed spaces, both real or both complex, and let $\dim X = n$ and $\dim Y = m$. Let $E \colon= ( e_1, \ldots, e_n )$ be an ordered basis for $X$, and let $F ...
3
votes
1answer
467 views

Equality in Young's inequality for convolution

I am trying to understand how sharp Young's inequality for convolution is. The inequality says $||f \ast g||_r \leq ||f||_p ||g||_q$ where as $1/p+1/q = 1+1/r$. Actually, there are a couple of papers ...
0
votes
1answer
34 views

Prob. 8, Sec. 4.5 in Kreyszig's functional analysis book: The inverse of the adjoint operator is the adjoint of the inverse operator

Let $X$ and $Y$ be normed spaces, both real or both complex, let $B(X,Y)$ denote the space of all the bounded linear operators $T \colon X \to Y$, and let $T^\times$ denote the adjoint operator of ...
2
votes
1answer
23 views

How to apply Theorem 4.3-3 in the proof of Theorem 4.5-2 in Kreyszig's functional analysis book?

Here's Theorem 4.3-3 in Introductory Functional Analysis With Applications by Erwine Kreyszig: Let $X$ be a normed space and let $x_0 \neq 0$ be any element of $X$. Then there exists a bounded ...
3
votes
1answer
355 views

Why is semi-norm special?

One difference between semi-norm and norm is: "It is possible for $\|v\| = 0$ for nonzero v, $\|\cdot\|$ being semi-norm" I see some papers, and they use semi-norm directly. Why is semi-norm ...
2
votes
2answers
49 views

Compact linear operators $S$ and $T$, show that $S(I-T)=I$ if and only if $(I-T)S=I$ and deduce that $I-(I-T)^{-1}$ is a compact operator

If $T:X\to X$ is a compact linear operator, then for any bounded linear operator $S:X\to X$ we have that $S(I-T)=I$ if and only if $(I-T)S=I$. Where $X$ is a normed space, also $T$ is bounded. With ...
2
votes
1answer
34 views

Manipulating sets ($+$, etc).

I was seeing a proof of the Open Mapping Theorem, in Kreyszig's book, and I have no problems with it. But there's a point in which he does something like: $$\begin{align}B_Y(0,r) \subset ...
0
votes
1answer
20 views

$p$-operator space property

If $S,T,U,V\in B(L_p(X,\mu))$, $p\in[1,\infty)$, and we regard $\begin{pmatrix} S & T \\ U & V \end{pmatrix}$ as an operator on $B(L_p\oplus_p L_p)$, then supposedly we have ...
0
votes
1answer
17 views

Comparing norms - $M_{kn}(A)$ versus $M_k(M_n(A))$

I am having to deal with the problem of passing between $M_{kn}(A)$ and $M_k(M_n(A))$ where $A$ is a Banach algebra, removing parentheses in one direction, and adding parentheses in the opposite ...
1
vote
2answers
36 views

Prove that the normed spaces $(C[0,1], \| \cdot\|_2)$ and $(C[a,b], \|\cdot \|_2)$ where $\| \cdot\|_2$ is the Euclidean norm are isometric.

Essentially I'm looking for a bijection $f: C[0,1]\to C[a,b]$ such that$$\|f(x) \|_2=\| x\|_2$$ I don't know how to go about finding this function, but I do know that it is possible. $$\| x \|_2 = ...
0
votes
1answer
30 views

Define a metric using scalar product and prove that it is indeed a metric

So this is how I went about this: $\langle\,\cdot\,,\,\cdot\,\rangle: X \times X \to \mathbb{R}$ such that (by definition I list the properties of scalar product) and I can east prove the first three ...
2
votes
1answer
18 views

Proof that every polilinear map who's domain is $R^{n_1} \times R^{n_2}… \times R^{n_k}$ and co-domain any given real normed space Y is bound.

A Polilinear map\operator is $P:X^1 \times ... \times X^n \to Y$ such that the foolowing applies: $\lambda, \mu \in R$ $$ P( \lambda x_1^1 + \mu x_2 ^1, x^2,...,x^n)= \lambda P(x_1^1,x^2,...x^n)+ \mu ...
0
votes
1answer
36 views

Can anyone help out with this proof, certain steps are unclear. Norm of linear operator.

I have the following norm defined as follows (in $R^n$, $x=(x^1,x^2,\ldots,x^n)\ $)$\| x\|_1= \sum_{i=1}^{n}|x^i|$ Let $A:R^m \to R^n$ a linear map of the spaces $(R^m ,\| \cdot \|_1 )$ and $((R^n ...
1
vote
1answer
16 views

Comparison of the norms of two non-negative real-valued vectors differing only in one component.

Let $0 \leq a \leq b$ and let $\mathbf{x} \in \mathbb{R^{n}}$. Let $\|.\|$ be a norm over $\mathbb{R}^{n+1}$. If we write $( \mathbf{x},a)$ the vector of $ \mathbb{R}^{n+1}$ made by concatening ...
3
votes
0answers
29 views

Operator Norm and Submultiplicativity against the Spectral Norm

Consider $\mathcal{A}:\mathbb{R}^{n\times m}\to \mathbb{R}^{p\times q}$ to be a linear operator. I know that by considering the trace norm and using the submultiplicativity of the operator norm we ...
0
votes
1answer
49 views

prove the continuity of $T_\phi f=\int_0^1 f(x)\phi(x) \,dx\\$ [duplicate]

Let $\phi\in C[0,1]$ and let $T_\phi~:C[0,1]\rightarrow\mathbb{R}$, defined as $T_\phi f=\int_0^1 f(x)\phi(x) \,dx\\$. How can i prove that it's a continuous operator?
0
votes
0answers
16 views

Bounded v.s. completely bounded homomorphisms between $L^p$ operator algebras

Take an $L^p$ operator algebra to mean a closed subalgebra $A\subset B(L^p(X,\mu))$ for some ("nice") measure space $(X,\mu)$, $p\in[1,\infty)$. Equip the matrix algebra $M_n(A)$ with the norm ...