-3
votes
1answer
35 views

Exercise 3.6: Elementary Functional Analysis By Barbara [on hold]

Let $X=\ell_\Bbb R^\infty$ denote the space of bounded sequences with real entries, in the supremumnorm. Consider the operator $T$ defined on $X$ by $T(x_1, x_2, . . .) = (x_2, x_3, . . .)$; this is ...
0
votes
1answer
42 views

Question on sequence space (as a linear space)

Let $X$ be the space $\ell_\infty$ of all bounded sequences of real scalars. If $Y$ is the set of all $x\in X$ that have bounded partial sums (1) Can I say $Y$ is a linear space (as a subspace of ...
1
vote
0answers
29 views

Question about Hahn-Banach separation Theorem

So here is my question, I am just reading about the Hahn-Banach separtion Theorem and there is one case where a question appeared, namely, Let $X$ be a normed $\mathbb R$ vectorspace and let $A,B$ ...
3
votes
3answers
52 views

Regarding a Basis for Infinite Dimensional Vector Spaces

In my linear algebra class, during the discussion of vector spaces, our instructor mentioned infinite dimensional spaces, including the polynomial space over Q and the space of all continuous ...
0
votes
1answer
14 views

is the space of finite sequences Frechet?

Let $E:=\coprod_{\mathbb{N}}\mathbb{R}$ be the space of all finite, real sequences equipped with the final structure wrt to all injections $inj_k(x)=(0,0,\dots,x,0,\dots)$. Since completeness of $E$ ...
0
votes
2answers
49 views

$l^r \subset l^p$ and is it even a subspace

It is true that for $r<p$ and $r,p \in [1,\infty)$ we have that $l^r \subset l^p$. Is it true that $l^r$ cannot be isomorphic to a subspace of $l^p$?
0
votes
0answers
19 views

Confusion with proving that some subspace of a Banach-Space is closed

So here is my problem, I am trying to show that, Let $X,Y$ be Banach-Spaces and $T:X\rightarrow Y$ a linear and bounded map. Then $T(X)$ is closed if $Y/T(X)$ is of finite dimension. While ...
5
votes
1answer
42 views

Is this condition sufficient to determine the linear space is of finite dimension?

From the Banach theory we knew that: 1) A linear space(a vector space endowed with its vector topology) $X$ of finite dimesion $dimX=n$ has the following property: If ${\left\| \bullet \right\|_1}$ ...
1
vote
0answers
29 views

Want to determin a dual space

I would like to determine the dualspace of some normed vectorspace. Namley, $$c_0:=\{x=(x_n)_{n\mathbb N}\subset\mathbb R:\lim_{n\rightarrow\infty}x_n=0\; \text{ and } ||x||=\sup_n|x_n|\}$$ I ...
2
votes
1answer
46 views

Is a normed $\mathbb R $ vectorspace complete in general?

I would like to finde out if a normed $\mathbb R$-Vectorspace is complete in general. Or even in a more general case if a normed $K$-Vectorpace, where K is a close field is complete? I somehow think ...
1
vote
1answer
49 views

Which of these things is not like the others?

What's in a name? Well quite a lot, if you're confused enough. I have an engineering-style mathematics education, based on good old hand waving and learning bits and pieces from all over the place. I ...
1
vote
1answer
61 views

Why each nonempty weakly open set of an infinite dimensional normed linear space is unbounded with respect to the norm

Suppose $V$ is an infinite dimensional vector space, $f_i$ ($i$ is from $1$ to $n$) are real-valued linear functions on $V$, I cannot understand why the intersection of kernels of $f_i$ must contain ...
0
votes
1answer
32 views

Prove that for every subspace we can find a finite number of linear functionals such that $W=\ker l_{1}\cap\cdots\cap \ker l_{k}$

In need of some assistance regarding this questions from a University textbook (I'm learning by myself). Its about Dual Spaces: Let there be $V$ a finite vector space (Has a basis) over $\mathbb{F}$. ...
0
votes
1answer
16 views

Help with Dual Spaces - Prove that either $w\in Im(f)$ **or** there exists ${l\in W^{*}}$ such that $f^{*}\left(l\right)=0$ **and** $l(w)=1$"

I'm in need of some assistance regarding this question. I'm learning Linear Algebra by myself using a university textbook and it has this question regarding Dual Spaces: "Let there be a linear map ...
4
votes
1answer
43 views

Duality mappings on finite-dimensional spaces

I have a few questions regarding some concepts in a book "nonlinear partial differential equations by Roubicek" I am studying. The following is from the text. "Let $V$ be a separable, reflexive ...
0
votes
2answers
19 views

Extending Continuous Basis

It is given $(k-d)$ continuous vector-valued functions $K_1,\dots,K_{k-d}:\mathbb{R}\mapsto\mathbb{R}^k$, with $d\leq k$. Suppose that for all $x\in\mathbb{R}^k$, the set ${\cal ...
1
vote
0answers
34 views

Getting “semi” orthogonal basis from a linear independent set

Let $K_i: \mathbb{R}\mapsto \mathbb{R}^k$ are continuous functions for all $i=1,\dots,k-d$ such that for every fixed $t\in\mathbb{R}$ we have ${\cal K}_t=\{K_1(t),\dots,K_{k-d}(t)\}$ be a linear ...
0
votes
1answer
23 views

Example for a two norms and vector space which are equivalent on this Vectorspace

I need an example where the two norms and the vector space are equivalent on the vector space.
0
votes
2answers
69 views

How to prove $l^3(\mathbb R)$ is a vector space.

I have a small confusion,i know how to prove $l^2(\mathbb R)$ is a vector space but i am not getting any idea to prove $l^3(\mathbb R)$ is a vector space. Vector space is defined as
2
votes
1answer
97 views

Linearly independent functionals

Let $ f_1,\ldots,f_n$ be linearly independent linear functionals on a vector space $X$. Show that there are $n$ elements $x_1,\ldots,x_n$ in $X$ such that the $n\times n$ matrix $[f_i(x_j) ]$ is ...
1
vote
0answers
40 views

Linearly independent linear functionals [duplicate]

Let $ f_1,\ldots,f_n$ be linearly independent linear functionals on a vector space $X$. Show that there are $n$ elements $x_1,\ldots,x_n$ in $X$ such that the $n\times n$ matrix $[f_i(x_j) ]$ is ...
6
votes
2answers
119 views

Why do we need dual space [closed]

In functional analysis there are many places where dual space is mentioned, but I still don't understand the real power of that concept. Why do we need the dual space?
2
votes
1answer
82 views

Direct sum of eigenspaces of a compact operator has finite codimension

In an infinite dimensional Hilbert space the orthogonal complement of the (closure) of the direct sum of eigenspaces of a compact normal operator is finite dimensional. Why is this the case? thanks.
1
vote
1answer
40 views

Orthogonal complement in pre-hilbert space

I just want to be sure that the following is correct: Let $T:H \rightarrow H$(continuous), where $H$ is a pre-Hilbert space, then we have $H=\ker(T) \oplus\ker(T)^{\perp}$, where $\ker(T)^{\perp}$ is ...
1
vote
2answers
51 views

Chart of how the mathematical spaces are related? (soft question)

When dealing with specific function spaces e.g. Sobolev, Hilbert, etc., I find it easy enough to accept the properties of that space and work with them; however, I have a hard time visualizing how ...
0
votes
0answers
17 views

Given data, approximations in a metric space for moving into a normed vector space isometrically.

Please see this question and this answer. Here $f_x(y)$ is approximated by $$x_v = [d(x,K_1),d(x,K_2),....d(x,K_N)]$$ by choosing to consider distances from $x$ to only certain points $K_i$ and ...
1
vote
1answer
38 views

Comparison of Symmetric Operators

The Problem: There is a unitary space $(V,<.,.>)$, $D \subseteq V $ a subspace and $ A,B : V \supseteq D\to V $ are two symmetric linear operators. Show that if: $<Ax , x> $$=$ $<Bx ...
3
votes
3answers
75 views

For closed sets, is $\text{cl}(A+B)=\text{cl}(\text{cl}(A)+\text{cl}(B))$?

Let $A$ and $B$ be nonempty subsets of $\mathbb{R}^n$, then is $\text{cl}(A+B)$ equal to $\text{cl}(\text{cl}(A)+\text{cl}(B))$? If that is true, then how to prove it? If they are not equal, then ...
0
votes
1answer
36 views

Proving Density of Subset of Hilbert Space

Suppose we have a subspace, $M$, of Hilbert space $H$. Prove the first statement implies the second statement: 1) If $<f,g> = 0$ for any $g\in M$, then $f=0$ in $H$. 2) $M$ is dense in $H$. I ...
0
votes
2answers
71 views

Compactness in Infinite Dimensional Vector Spaces

Show that, in an infinite dimensional normed space $(V,\|\cdot\|)$, the closed ball of radius $2$ $$ B_2:=\{x\in V:\ \|x\|\leq2\} $$ is not compact. I suspect I am not understanding what is going ...
2
votes
0answers
60 views

Linear Operators: Continuous $\Rightarrow$ Bounded

Let $T:V\rightarrow V'$ be a continuous linear operator between two normed vector spaces $V,V'$. Show that it is bounded. Continuity is defined as $\lim_{n}\|x_n-x\|=0\Rightarrow ...
6
votes
1answer
230 views

Vectors, Basis, Dual Vectors, Dual Basis and Tensors

I'm trying to understand tensors and I know they have something to do with the basis and the dual basis of a vector space and a dual space. First I will give a concrete example to make clear what I ...
1
vote
1answer
40 views

Extension of a zero linear functional

How can I show using Hahn-Banach theorem that, if $E$ is a real vector space, $F$ is a proper vector subspace of $E$, and f is the zero linear functional $f:F\to\mathbb{R}$ such that $f(x)=0$ $\forall ...
3
votes
1answer
77 views

Is there always an injective map from a space in its dual space?

Today our teacher said that dual spaces are "big" and told us that this is a consequence by Hahn-Banach's theorem. So I was wondering whether the dual space of a space is always "bigger" or equal ...
3
votes
1answer
84 views

Subspaces of a Topological Vector Spaces

I have a few questions about topological spaces which I am currently studying. First some definitions that I am using: Definition of subspace topology: Given a topological space $(X,\tau)$ and a ...
1
vote
1answer
60 views

Prove this is a bounded linear operator and find its operator norm?

I have a map $$A:(C[0,1], || \centerdot ||_\infty) \rightarrow \mathbb R, Ax = x(0) \forall x \in C[0,1]$$ and need to prove it's a bounded linear operator, and find its operator norm. I've tried ...
1
vote
2answers
38 views

$X$ be a normed space and assume that $E \subset X$ such that $\operatorname{int}(E) \neq\varnothing$

Let $X$ be a normed space and assume that $E \subset X$ such that $\operatorname{int}(E) \neq \varnothing$ then show that $E$ spans $X$. I am trying it in a following way.... Let be the norm ...
0
votes
1answer
92 views

$l_0$ is all sequences with finitely many non-zero terms. Show $W^\perp=\{y: <x,y>=0, x\in W\}=\{0\}$ where $W = \{x : <x,a>=0\}$.

Consider the inner product space $l_0$ consisting of all infinite sequences of complex numbers with only finitely many non-zero terms, with the inner product of $l^2$ (space of square summable ...
2
votes
2answers
58 views

$T$ be the operator from $C[0,1]$ to $C[0,1]$ defined by $Tf = f'+f''$. Show that the operator $T$ is unbounded.

$f \in C[0,1]$, the space of all continuous, complex-valued functions on $[0,1]$ with supremum norm. $\|f\|=\sup_{x\in[0,1]}|f(x)|$. Let $D$ be the set of $f \in C[0,1]$ such that the first ...
2
votes
1answer
121 views

Functional analysis, help in Hahn-Banach theorem application

$M$ is the subspace of $L_p[a,b]$ that $\forall f\in L_p[a,b]$ $\exists g\in M$ with $f(t)\leq g(t)$ almost everywhere. $T:M\rightarrow \mathbb{R}$ $\quad$$T(f)\geq0$ whennever $f(t)\geq0$ a.e ...
2
votes
1answer
77 views

How to prove $a_1^Ta_1+a_2^Ta_2\le b_1^Tb_1+b_2^Tb_2.$

Let $p,q > 0$, $a_1, b_1\in \mathbb{R}^m, a_2,b_2\in\mathbb{R}^n$ be vectors. Given that ...
2
votes
1answer
107 views

Could we write Fourier transform as a matrix?

I have heard that Fourier transform is a linear transformation. I have also heard that any linear transformation can be written as a matrix multiplication. (probably I'm missing some details in the ...
4
votes
2answers
106 views

Show that $F+G$ is closed when $G$ a closed subspace of normed space $E$ and $F$ a finite dimensional subspace of $E$.

Question: Let $E$ be a normed space. Let $G$ be a closed subspace of $E$ and let $F$ be a finite dimensional subspace of $E$. Show that $F+G$ is a subspace of $E$ and is closed. I'm having trouble in ...
2
votes
1answer
89 views

If A and B are compact than also A+B

Suppose we have a topological vector space $X$ and $A, B\subset X$. We define A+B to be the set of the sums $a+b$ where $a\in A$ and $b\in B$. We should prove that also A+B is compact if A and B are ...
-1
votes
1answer
73 views

proving boundedness of a linear functional

Let $f$ be a linear functional on a normed vector space V and $f^{-1}(\{0\})$ is closed. Prove that $f$ is bounded
0
votes
1answer
234 views

closed subspace of normed vector space

Is every finite dimensional subspace of a normed vector space closed? If yes, please prove it or else give a counter example.
5
votes
5answers
2k views

What are some examples of infinite dimensional vector spaces?

I would like to have some examples of infinite dimensional vector spaces that help me to break my habit of thinking of $R^n$ when thinking about vector spaces.
1
vote
0answers
50 views

Can you construct a coutable local base in the space of continuous functions?

Let $(C,\tau)$ be the topological vector space of all complex continuous functions on $[0,1]$ with seminorms $p_x(f)=|f(x)|$, $x\in [0,1]$. We have known $(C,\tau)$ is not metrizable,but how could I ...
1
vote
0answers
47 views

Does a limit exist in a non-metrizable space?

Let $C$ be the vector space of all complex continuous functions on $[0,1]$. And let $(C,\tau)$ be the topological vector space defined by the seminorms $$p_{x}(f)=|f(x)|~~~~(0\leq x\leq 1).$$ We have ...
0
votes
1answer
86 views

Is a subspace closed?

Let $X$ be a topological vector space,$N$ a subspace.Is $N$ closed in the topology of $X$ if $X/N$ is closed in its suitable toplogy?\par Note that every singleton of $X$ is a closed set,so I think ...