1
vote
0answers
16 views

Strict convexity and best approximations

Let $V$ be a normed vector space. It is said to be strictly convex if its unit sphere does not contain nontrivial segments. A subset $A \subset V$ is said to have the unicity property if for any $x ...
3
votes
1answer
45 views

Is the set of all Taylor polynomials a vector space?

Let $V$ denote the set of all Taylor polynomials of degree $\leq n$ for a fixed natural number $n$ (including the zero polynomial), regraded as real-valued functions of a real variable. Then is $V$ a ...
2
votes
1answer
50 views

Series in a space which is not complete

Let $X$ be a normed vector space and $\left\lbrace x_n \right\rbrace_{n \in \mathbb{N}} \in X^{\mathbb{N}}$ with $$\sum_{n=1}^{\infty} \|x_n\| < \infty \wedge \sum_{n=1}^{\infty} x_n \notin X,$$ ...
1
vote
1answer
44 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 ...
4
votes
1answer
53 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 ...
1
vote
1answer
36 views

Left shift operator $L: l^2 \rightarrow l^2$ on the sequence space $l^2$

$$L: l^2 \rightarrow l^2$$ is defined by $$b = (b_1,b_2,...) \mapsto Lb = (b_2,b_3,...)$$. $(Lb)_n = b_{n+1}$ respectively. How can I determine the adjoint endomorphism $L^*$? Kind regards George
2
votes
1answer
35 views

Difference between F-space and Frechet space in W. Rudin's “Functional Analysis”

In Walter Rudin's book, "Functional Analysis", we read that by talking about local base, he will be thinking about neighborhoods of $0$. In the vector space context, the term local base will ...
2
votes
1answer
49 views

Linear Functional: Continuous? [duplicate]

Given a Banach space: $E$ and chosen a Hamel basis: $\mathcal{B}$ Any vector induces a (noncanonical) algebraic linear functional by: $$\delta:E\to E^*:\delta_b(b'):=\delta_{b,b'}\text{ defined ...
0
votes
0answers
23 views

Show that $\{w^{1/2}\phi_n\}$ is an orthonormal set in $L^2(D)$ if $\{\phi_n\}$ is an orthonormal set in $L^2_w(D)$

As mentioned in the title, my problem is: Show that $\{w^{1/2}\phi_n\}$ is an orthonormal set in $L^2(D)$ if $\{\phi_n\}$ is an orthonormal set in $L^2_w(D).$ So I know that: ...
3
votes
0answers
41 views

Vector Space Verification

I just took an exam asking me if the following are a vector space over $\mathbb{R}$ assuming that the set of all real valued functions on the interval $[0,1]$ is a vector space with theoperations ...
0
votes
0answers
15 views

Sufficient conditions for RTree

What is the sufficient screening criteria of a space for the possibility to use R-Tree spatial index on it? I cannot apply it to a space with just Jaccard distance as the metric. As I suppose the ...
0
votes
0answers
37 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 ...
4
votes
0answers
98 views

Do infinite dimension vectors hold the same properties as finite dimension vectors?

I am learning about vectors and am wondering if finite dimension vector operations such as the dot product hold for infinite dimensional vectors?
0
votes
0answers
52 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 ...
5
votes
1answer
79 views

Is it possible to define an inner product such that an arbitrary operator is self adjoint?

Given a vector space $V$ (possibly infinite dimensional) with inner product $(.,.)$. We say an operator $A$ is self adjoint if $(Af,g)=(f,Ag)$. The definition as stated require us to start with an ...
0
votes
3answers
30 views

Some questions of vectors and dense subsets

I have a couple of quick functional analysis related questions: 1.Say we have a normed space $V$ and reflexive, separable Banach space and $K \subset V$ a closed, convex, bounded subset of $V$. ...
1
vote
3answers
40 views

$C^0([a,b])$ is an infinite dimensional vector space

I am proving that $C^0([a,b])$ is an infinite dimensional vector space. The fact that it is a vector space is clear. But I cannot understand how to prove that it has infinite dimension. Let ...
3
votes
1answer
73 views

Does existence of a non-continuous linear functional depend on Axiom of Choice?

Well, it is easy to construct a non-continuous linear functional on an arbitrary infinite-dimensional vector space (assuming Choice, and taking a basis etc.). I think it is intuitive to say that: ...
1
vote
0answers
22 views

Simultaneous extension and complemented subspace

The following is Exercise 3.13.5 of Conway's Functional Analysis: Let $X$ be a compact set and let $Y$ be a closed subset of $X$. A simultaneous extension for $Y$ is a bounded linear map $T:C(Y)\to ...
1
vote
0answers
29 views

Linear map in Hilbert space.

If you have a linear map $h\mapsto T(h)$ from $H_1$ a real separable space, to Hilbert space $H_2$, it seem that this maps provides an isometry of $H_1$ onto a closed subspace of $H_2$. I try to ...
2
votes
3answers
48 views

Prove that if $\dim X'<\infty$ then $\dim X<\infty$

I have to prove that $\dim X'<\infty$ then $\dim X<\infty$ where $X$ is a normed vector space and $X'$ is a space of all linear and continuous functionals from $X$. How can I prove this? I ...
3
votes
1answer
51 views

How to show: $\exists$ a subspace $L$ of a Hilbert space $H$ such that $H\neq L\oplus L^{\perp}$.

How to show: $\exists$ a subspace $L$ of a Hilbert space $H$ such that $H\neq L\oplus L^{\perp}$. Let consider $H=l_2$ where $l_2=\lbrace x=(x_n)^\infty_1: \sum^\infty_1 |x_n|^2<\infty \rbrace $ ...
0
votes
1answer
66 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
37 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
82 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
18 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
50 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
25 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
31 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
47 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
61 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
67 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
50 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
25 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 ...
5
votes
1answer
63 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
23 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
39 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
28 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
76 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
115 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
152 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
94 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
58 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
55 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
39 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
78 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
51 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 ...