Functional analysis is the study of infinite-dimensional vector spaces, often with additional structures (inner product, norm, topology), with typical examples given by function spaces. The subject also includes the study of linear and non-linear operators on these spaces, as well as measure, ...

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1answer
31 views

Space of complex poynomials

Let $\mathbb{C}_n[z]$ be the space of polynomials (of degree $\le n$) with complex coefficients, let the inner product be $(p,q):=\int_{-1}^1p(t)\overline{q(t)}dt$. There is one and only one $K_{w} ...
2
votes
2answers
79 views

$C^\infty$ approximations of $f(r) = |r|^{m-1}r$

Consider $f(r) = |r|^{m-1}r$ where $m \geq 1$. Is it possible to find $C^\infty$ functions $f_n$, such that $f_n \to f$ uniformly on compact subsets of $\mathbb{R}$, $f_n' \to f'$ uniformly on ...
2
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1answer
926 views

Proof of the Unsöld's Theorem (the sum of spherical harmonics)

There is an identity concerning spherical harmonics that plays a pretty important role in atomic physics. Thanks to wikipedia (http://en.wikipedia.org/wiki/Spherical_harmonic) I know that its name is ...
2
votes
1answer
96 views

Trace Operator on $L^2$ Functions

in Why no trace operator in $L^2$? you mentioned, that there exists a linear continuous trace operator from $L^2(\Omega)$ to $H^\frac12(\partial\Omega)$* for sufficiently smooth boundary. Can you give ...
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2answers
65 views

Cauchy-Schwarz inequality for symmetric operators

Suppose that A is a symmetric operator such that $(Au,u)\geq 0$ where $(\cdot,\cdot)$ denotes the inner product. How do I show that $|(Au,v)|\le (Au,u)^{1/2} (Av,v)^{1/2}$? I can't figure out how the ...
6
votes
2answers
172 views

Showing the set $A+B$ is closed.

Let $X$ be a banach space, and let $A$, and $B$ be closed linear subspaces. Assume that $$\inf\{\|x-y\|\mid x\in A, y\in B, \|x\|=\|y\|=1\}>0$$ I want to show that $A+B$ is closed. I was ...
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0answers
48 views

Linear homeomorphism implies a Schauder basis

Problem: Let S be a Schauder basis of H. Let L:H $\rightarrow$ H be a linear homeomorphism. Show that L(S) is a Schauder basis. L is a linear homeomorphism if it is one-to-one, linear, continuous, ...
2
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0answers
40 views

Topological modules with enough continuous linear functionals.

Context: I'm trying to find out which topological (unital) modules are "good enough" for generalizing results from real or complex functional analysis. For example, I say that a module, in order to be ...
1
vote
1answer
62 views

Question concerning weak convergence

So here is my problem, I want to show the following, Let $X$ be a normed $\mathbb K$-Vectorspace. And let $(x_n)_{n\in\mathbb N}\subset X$ be such that it converges weakly to some $x\in X$. Then, ...
0
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1answer
93 views

Unit ball weakly* compact

So here is my problem, I am trying to understand the proof of, $X$ Banach space $\Rightarrow$ the unit ball in $X^*$ is weakly* compact. The proof uses Tychonoffs Theorem to conclude the ...
0
votes
1answer
24 views

The index of some perturbation about elliptic operator with Robin boundary condition

Let $I$ be an closed interval $[0, 1]$. $C^{2}(\bar{I})$ is the space of all $C^{2}$ functions on $(0, 1)$ with continuity at boundary and usual maximal norm. $C(\bar{I})$ is the space of all ...
2
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0answers
55 views

Comparison of two final topologies

Consider the vector space $F$ of all infinite sequences of reals numbers, such that only finitely many terms of each sequence are nonzero. I recently encountered an exercise where I was required to ...
2
votes
0answers
195 views

Newton boundary condition for second order pde

I have a few questions about Newton boundary conditions for a second-order partial differential equation: $$-\text{div}(a(x,u,\nabla u)) + c(x,u,\nabla u)$$ considered on a bounded connected ...
1
vote
1answer
119 views

Need help understanding this proof about Gelfand spectrum

Consider the following theorem: Let $A$ be a complex non-unital commutative Banach algebra and let $\Omega (A)$ denote its Gelfand spectrum / character space. Then $\Omega (A)$ is locally compact. ...
3
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0answers
78 views

$L^2(S;W^{1,p})$-regularity for solution of parabolic pde in Hilbert space setting

I have a question regarding the regularity theory for parabolic pdes. I am considering the following "minimal working example": Let $\Omega\subset\mathbb{R}^n$, $n\in\mathbb{N}$, be and bounded ...
1
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0answers
66 views

Equivalent to not separable?

I wanted to ask if this statement is true, Let $X$ be a normed vectorspace. Then, X is separable iff every disjoint, and open familiy of subsets is countable. My idea to prove it was the ...
1
vote
1answer
244 views

How can I show the statement: “every Cauchy sequence converges” can replace the completeness axiom?

I saw a theorem in my textbook that they claim the statement: "every Cauchy sequence converges" can replace the completeness axiom (the fact that every bounded sequence has a least upper bound). In ...
0
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1answer
59 views

How to show that $(C^0((a,b)), d_\infty)$ is not a metric space

Let $d_\infty:C^0([a,b]) \times C^0([a,b]) \to [0,\infty)$ be defined as $$ d_\infty(f,g)=\sup\limits_{x \in [a,b]} \left\{ |f(x) - g(x)| \right\} $$ I have already shown that $(C^0([a,b]), ...
0
votes
1answer
47 views

show that $(x_k)$ is convergent and limit$\notin\ell^p$

Let $\ell^p:=\{(x_n)|\|x_n\|_{\ell^p}<\infty\}$ be the space of sequences with finite $\ell^p$-norm. Show that the sequence $x_1:=1$, $x_k:=\frac{1}{\log k}$, $k\geq 2$ converges to $x=0$, but ...
0
votes
2answers
34 views

Looking for an example regarding hilbert spaces

Does there exists a hilbert space $\mathcal{H}$ such that a sequence $\{ x_n \} \subset \mathcal{H} $ is weakly convergent, but not convergent in the hilbert space norm ?
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3answers
371 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 ...
1
vote
1answer
76 views

Use Gram-Schmidt orthonormalization to find the first 4 terms of the orthonormal sequence obtained from…

Use Gram-Schmidt orthonormalization to find the first 4 terms of the orthonormal sequence obtained from $S=({x^n})_{n \epsilon N}$ in $L^2(0,1)=(\int^1_0 f^2(x)dx)^\frac{1}{2}$
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1answer
57 views

$x$ is orthogonal to $y$ iff $\|x+ay\| \geq \|x\|$ where $x \in C$

Show that in an inner product space, $x$ is orthogonal to $y$ iff $\|x+ay\| \geq \|x\|$ where $x \in C$. Proof: LHS: If $x$ is orthogonal to $y$, then $\langle x,y\rangle =0$. Let $a \in \mathbb C$. ...
2
votes
1answer
97 views

I would like to show that $\ell^1$ is separable

So here is my question, I want to prove that $\ell^1$ is separable. So i need to show that there exists a countable dense subset in $\ell^1$. Since I am not sure if my idea was right i hoped ...
1
vote
1answer
38 views

Is this norm induced by an inner product?

Is $||x||=\sqrt{\sum_{n=2}^{\infty} \frac{1}{n} x^2_n} +|x_1|$ induced by an inner product? I claim that this is not true because in a previous question I found that the inner product $<x,y>= ...
0
votes
1answer
86 views

What two continuous functions from $[a,b]\to \mathbb C$ failed the parallelogram rule?

Let $X=C([a,b],\mathbb C)$ with $\|\cdot\|_\infty$. Show that $\|\cdot\|_\infty$ is not induced by an inner product. Hence to show that this norm is not induced by an inner product we must prove that ...
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0answers
59 views

Transfer vector space properties to dual space

I am curious about this here (Actually, I don't know if my assumptions are true or not) a) Let $X$ be a Banach space that is isomorphic to $Y$, then $X^*$ is also isomorphic to $Y^*$. I sketched a ...
2
votes
2answers
63 views

How to find maximum/minimum of $y=\frac{x(x^2-x+2)}{x^2-9}$?

How can I find minimum and maximum of $y=\frac{x(x^2-x+2)}{x^2-9}$? In other words, points, where $y'=0$. My current steps are: $$y=\frac{x(x^2-x+2)}{x^2-9}=x-1+\frac{11x-9}{x^2-9}$$ ...
0
votes
0answers
79 views

Examples of topologies between norm and weak star

Let $X$ be a normed vector space and $X^\ast$ denote its continuous dual. The norm on $X^\ast$ is given by $\|\varphi\|=\sup_{\|x\|=1}|\varphi(x)|$. The weak star topology on $X^\ast$ is the weakest ...
0
votes
1answer
59 views

Prove that metric space is complete

I have metric space: $$ X = <[0,+\infty), \rho>, \rho(x,y) = |ln(1+x) - ln(1+y)|$$ I know it is complete, but I don't know how to prove it. How can I prove that fact?
3
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0answers
133 views

Banach Alaoglu different proofs

While trying to prove Banach-Alaoglu theorem I noticed the differrent equivalent definitions of compactness. When I tried to find a proof of Banach-Alaoglu I found a proof in Pedersen Analysis Now and ...
1
vote
1answer
67 views

Check equicontinuity of functions family

I have this family of functions: $$M = \{x(t) = (t-1)^{2n}, n\in\mathbb Z\}, t \in [0,1]$$ What should I do to check equicontinuity?
1
vote
1answer
45 views

Squares of C*-algebras

I'm reading a paper where it is claimed that every C*-algebra $A$ satisfies $A^2 = A$, "for example, using Cohen's 1959 factorization theorem". However, I don't see how to apply Cohen's factorization ...
0
votes
1answer
475 views

weak convergence implies point-wise convergence?

If we have a bounded sequence $\{f_n\} \in L^p[a,b]$ that converges weakly to $f$ does this mean that the converges is also pointwise?? thank you.
2
votes
1answer
69 views

Why are function spaces typically defined on open sets?

I never really bothered to ask this question and now it seems silly...but why do we always seem to define function spaces $X(U)$ (e.g. $L^2(U), BV(U)$ for open sets $U\subset\Bbb{R}^n$? What breaks if ...
0
votes
1answer
31 views

Question about functional analysis

LEt $X$ be an inner product space. $z \in X$ fixed. Let $f(x) = <x,z>$. Suppose the mapping $X \to X' $ given by $z \to f $ is surjective. Does it follow that $X$ is a Hilbert space? $X'$ is ...
1
vote
0answers
59 views

If the quotient of a subspace of a banach space is finite, is it a closed subspace?

Given a Banach space B,V is a subspace of B,if B/V is finite dimension,then is it enough to show that B is closed? Thanks!
1
vote
1answer
62 views

Non-locally convex topologies on $\mathbb{R}^{n}$ compatible with the vector space structure

So I know that every locally convex topology on $\mathbb{R}^{n}$ is equivalent to the norm topology. Are there any non-trivial examples of non-locally convex topologies on $\mathbb{R}^{n}$ that still ...
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vote
0answers
41 views

Showing a particular integral operator is trace class

Let $f$ and $P$ be continuous, integrable functions $\mathbb{R} \to \mathbb{C}$ vanishing at $\pm \infty%$. Concisely, $f,P \in C_0(\mathbb{R}) \cap L^1(\mathbb{R})$. Also, assume that $P$ is ...
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vote
2answers
62 views

showing $\inf \sigma (T) \leq \mu \leq \sup \sigma (T)$, where $\mu \in V(T)$

I am trying to prove the following: Let $H$ be a Hilbert space, and $T\in B(H)$ be a self-adjoint operator. Then for all $\mu \in V(T)$, $\inf \left\{\lambda: \lambda \in \sigma (T) \right\}\leq \mu ...
0
votes
1answer
25 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 $E$ ...
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vote
1answer
188 views

Banach-Alaoglu theorem and coarseness of weak star topology

Let $X$ be a normed space and let $X^\ast$ denote its continuous dual. There is a norm on $X^\ast$ defined by $\|\varphi\|=\sup_{\|x\|=1}|\varphi(x)|$. The weak star topology on $X^\ast$ is defined to ...
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votes
2answers
683 views

Any good, undergraduate level introductions to Functional Analysis?

In my lower division math classes, my instructors referenced functional analysis as essentially the extension of linear algebra to infinite dimensional vector spaces along with some real analysis. As ...
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vote
1answer
24 views

Integral's limit

Let $X$ be a Banach space and $A$ is a linear bounded operator on $X$. It is well known that for $|\lambda|> \|A\|,$ we have $$\|(\lambda I - A)^{-1}\| \leq \frac{1}{|\lambda|-\|A\|}.$$ Now, let ...
0
votes
2answers
113 views

Is this a basis for the dual space?

There is an example on Wikipedia that I don't understand and I'd appreciate some help. They define $\mathbb R^\infty$ to be the space of all sequences that are zero except for finitely many indexes. ...
2
votes
1answer
217 views

The image of an bounded and linear operator between two banach spaces is closed if the image is finite codimension?

So here is my problem, I would like to prove the following, Let $X,Y$ be Banach-Spaces and $T:X\rightarrow Y$ a linear and bounded operator. Then $TX$ is closed if it is of finite codimension i.e ...
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0answers
84 views

the dual space of $C([0,1])$

I'm studying Conway's Functional Analysis by myself. the following question is one of this book's Exercise. If $n\geq 1$ , does there exist a measure $\mu$ on $[0,1]$ such that $\int pd\mu =p(0)$ for ...
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vote
2answers
246 views

set of symmetric positive definite matrix open?

I consider a collection of symmetric positive definite matrices of the same dimension. I've learned it's an open set but have no clue about the proof. Also, can the symmetry condition be dropped? ...
0
votes
1answer
69 views

Finding measure given by Riesz Representation Theorem

I can show that if $K \subset \mathbb{C}$ is compact, then for any $a\in K,$ there exists a probability measure $\mu$ supported on $\partial K$ such that $f(a)=\int f\, \text{d}\mu$ for any $f\in A(K) ...
6
votes
1answer
89 views

Incorrect proof of Hahn Banach Theorem

What is wrong with the following trivial proof of the Hahn Banach Theorem? Hahn Banach Theorem: Let $V$ is a real normed vector space and $U$ a subspace. Then if $\phi : U \rightarrow \mathbb{R}$ is ...