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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19 views

Mean value of a function over the n-sphere superficie.

We know that we can use the bloch sphere to represent an unitary vectors $v$ in $\mathbb{C}^{2}$, due to the fact $su(2) \approx so(3)$. Then, if we have the function $f:\mathbb{C}^{2} \rightarrow ...
0
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0answers
20 views

$\{Q^{(n)}\}$ is tight? [on hold]

Let $Q^{(n)}$ , $n\ge 1$, are probability meaures in $E^{Q_+}$, where $E\subset \mathbb{R}^d$ ($d\ge 1$) is compact, $Q_+=Q\cap [0,\infty )$ and $E^{Q_+}$ denotes the collection of function from $Q_+$ ...
2
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2answers
30 views

Linear Projections: Bounded/Continuous?

Are linear (nonorthogonal) projections on (pre) Hilbert spaces necessarily bounded/continuous? (can you give a proof or counterexample)
1
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1answer
22 views

Are the continuous linear functions from a norm space to R bounded?

$\{X, \|\cdot\|\}$ a normed space, a function, maping from $X$ to $\mathbb R$, is linear and continuous. Is it a bounded linear function?
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1answer
59 views

Isometry from $\ell^1$ to $\ell^\infty$

Is there $f:\ell^1\to \ell^\infty$ so that $f$ is surjective $\forall x,y\in \ell ^1, \|x-y\|_1=\|f(x)-f(y)\|_\infty$
1
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0answers
23 views

Question about Schauder basis

The question is : Let $B$ be a Banach space and suppose $\{x_n\}$ the Schauder basis and $M$ be the space of sequence of scalars $\{a_n\}$ such that the sup norm of power series of $a_n x_n$ ...
1
vote
1answer
30 views

prove $\lim_{n\to\infty}\sup\{|f(x) - f_n(x) | : x \in S \} =0$

I understand how the theorem works but how would you prove that a sequence $f_n$ of functions on set $S \subset \mathbb{R}$ converges uniformly iff $$\lim_{n\to\infty} \sup\{|f(x) - f_n(x) | : x \in ...
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0answers
17 views

Dose the closed unit ball of C(the closure of E) with sub-norm, have no extreme points?

Let E be a bounded closed set in R^n. Dose the closed unit ball of C(E) with sup-norm, have no extreme points?
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1answer
30 views

German and French word for Basic sequence

Wikipedia says that a sequence $(x_n)$ is a basic sequence iff it is a Schauder basis of its closed linear span. I was wondering whether there is a French or German word for 'basic sequence'? How ...
2
votes
1answer
38 views

Is this space complete?

Let $X$ be the space of measurable functions $f:[0,1] \rightarrow \mathbb{R}$. I want to find out whether this space is complete under the metric $d(f,g):= \int_{[0,1]} \frac{|f-g|}{1 + |f-g|}$. Does ...
0
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1answer
17 views

Is $\phi(h)=\displaystyle \sup_{x\in X}{\|h-x\|}$ continuous on Hilbert space when $X$ is bounded?

Let $H$ be a hilbert space, and let $X \subset H$ be a bounded subset of $H$. Let define the function $\phi:H \to H$ by the rule $\phi(h)=\displaystyle \sup_{x\in X}{\|h-x\|}$. I want to know if this ...
0
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1answer
14 views

About a proposition of Willem's book

Let $\Omega$ be an open subset of $\mathbb{R}^N$ and let $2<p<\infty$. The functionals $\Psi(u)= \int_{\Omega} |u|^p , \chi (u)= \int_{\Omega} |u^+|^p$ are of class $C^2(L^p(\Omega), ...
1
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0answers
34 views

Proof that these are Fourier coefficients

I proved that for $f \in \ell^1 (\mathbb Z)$ its Gelfand transform $\widehat{f}$ is a map $\widehat{f}: S^1 \to S^1$ defined by $$ \widehat{f}(z) = \sum_{n \in \mathbb Z}f(n) z^n$$ In Murphy's book ...
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1answer
26 views

First resolvent equation

$(B,||\cdot||)$: banach space A family $(G_{\alpha})_{\alpha>0}$ of linear operators on $B$ with $D(G_{\alpha})=B$ for all $\alpha>0$ is called a strongly continuous contraction resolvent if ...
0
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1answer
22 views

Fixed Point Iterations for Bounded Affine Functions

Let $f: X \rightarrow X$ be continuous and $X \subset \mathbb{R}^n$ compact and convex. From Brouwer fixed-point theorem, $f$ admits a fixed point ($\bar{x} \in X$ such that $f(\bar{x}) = \bar{x}$). ...
4
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2answers
74 views

On the convexity of a particular form of integrals

EDIT: I made some critical corrections below. Let $\mathcal{H}\colon\mathbf{w}\cdot\mathbf{x}+c=0$ be a hypeplane in $\mathbb{R}^n$. Also, let $g\colon\mathbb{R}^n\to\mathbb{R}_+$, be a non-negative, ...
1
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0answers
31 views

Norm of a projection in $L_p$

Let $\mu$ be a probability measure on $[0,1]$ and let us define the projection $P$ on $L^p(\mu)$: $$P \; \colon f \mapsto \mathbb{E}(f){\bf 1}$$ (where ${\bf 1}$ is the constant 1 function). What is ...
0
votes
1answer
46 views

Gelfand transform is a bijection between $\ell^1$ and $\mathbb D$?

Let $A=\ell^1 (\mathbb Z)$. I read that it is possible to identify $S^1$ with the character space $ \Omega (A)$. But I have constructed a proof that identifies $ \Omega (A)$ with $\mathbb D$, the ...
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votes
1answer
21 views

Use Gram-Schmidt orthonormalization for $l^2$-space [closed]

Use Gram-Schmidt orthonormalization to find the first 4 terms of the orthonormal sequence obtained from $S=({a_n})_{n \epsilon N}$ in $a_j=(1,1/2,...,1/j,0,...)$ in $l^2=(a_j \epsilon R$ where ...
0
votes
1answer
40 views

Operator between two Hilbert spaces that preserve inner product must be linear

The Question is: If $M$ and $N$ are Hilbert spaces and $U : M \to N$ is a surjective function such that $\langle Uf ,Ug \rangle = \langle f, g \rangle$ for all vectors $f$ and $g$ in $M$, then $U$ is ...
0
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1answer
34 views

Cardinality of the Set of $\mathbb{C}$ valued sequences

Working a functional analysis question that I believe requires this and I'm struggling to determine this set's cardinality".
2
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1answer
30 views

Alternating projections on a Hilbert space

Let $P_1, P_2$ be the orthoprojections onto $S_1, S_2$, closed subspaces of a Hilbert space $H$. It is straightforward to show that if $(P_1P_2)^nx \to z$ then $z \in S_1 \cap S_2$ (I can post a quick ...
2
votes
1answer
32 views

Equicontinuous family of sequence of functions

We are given a sequence of real valued functions $\{g_n\}$ that are defined and continuous on the unit sphere $S$ and differentiable inside it (except at the boundary of the sphere $S$ Also, it is ...
1
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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 $A,B$ ...
0
votes
1answer
21 views

If $x_n \to x$ in $L^1(X,H)$ then $\langle x_n, h \rangle \to \langle x, h \rangle$ in $L^1(X,\mathbb{R})$

Suppose $H$ is a Hilbert space. Is it true that if $x_n \to x$ in $L^1(X,H)$ then $\langle x_n, h \rangle \to \langle x, h \rangle$ in $L^1(X,\mathbb{R})$ for any fixed $h\in H$? Certainly if $x_n\to ...
1
vote
0answers
108 views

Is this sequence a Cauchy sequence?

Consider a continuous mapping $f: X \rightarrow X$, where $X \subset \mathbb{R}^n$ is compact and convex. Consider a sequence $\{x_k \in X\}_{k \geq 0}$ such that for all $k \geq 0$ and $h \geq 1$, ...
0
votes
1answer
31 views

Convolution of a continuous function and uniform continuity

Suppose $f$ is a continuous function, and let the convolution $f_n(x) := f \star \varphi_n(x)$ where $\varphi_n$ are smooth test functions. We know $f_n \in C^\infty$. We know that if $f$ is ...
0
votes
1answer
25 views

In Inner Product Space ( not complete), dose a closed linear subspace equal to the the orthogonal complement of its orthogonal complement? [duplicate]

It is apparently that this holds in Hilbert space, but I can not prove this for general inner product space or find a counterexample. (The only not complete inner product space known to me is $L^2$ ...
2
votes
1answer
52 views

is this true for Hilbert space direct sum of $H$ when $H$ is infinite dimensional?

Let $(H_{\alpha})_{{\alpha \in I}}$ be a $I-$indexed family of Hilbert spaces over $\mathbb{F}$. let $H=\bigoplus H_\alpha$ be their Hilbert space direct sum. Can we say $\dim ...
3
votes
0answers
58 views

Rising Sun Inequality (Dunford-Schwartz maximal inequality)

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be an absolutely integrable function, and let $f^*:\mathbb{R} \rightarrow \mathbb{R}$ be the one-sided signed Hardy-Littlewood maximal function $$f^*(x) := ...
1
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0answers
21 views

Examples of semigroups of contractive Fourier multipliers but not positive?

Can you show me a concrete an example of semigroup $(T_t)_{t\geq 0}$ of Fourier multipliers such that each operator $T_t$ induces a contractive Fourier multiplier $T_t\colon L^p(\mathbb{T}) \to ...
1
vote
1answer
52 views

On weak convergence

I have the following Statement to prove Let $C$ be a closed, bounded and convex subset of a $\mathbb{K}$-Vectorspace $X$. Define a Support function $S_C:X^*\rightarrow\mathbb{R}$, $f\rightarrow ...
1
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0answers
32 views

Weak sequential compactness in a reflexive space

Let $\{X, \| \cdot \|\}$ be a normed space, $B$ is the unit ball of $X$. If $\{X, \| \cdot \|\}$ is reflexive, then is $B$ weakly sequentially compact? If it's not true, are there any counterexamples ...
1
vote
1answer
54 views

Using the topology of uniform convergence for functions over non-compact spaces

Let $(X, d)$ be a (complete) metric space, and $C(X)$ be the space of continuous maps over $X$. If $X$ is compact, one often uses the topology of uniform convergence when analyzing $C(X)$. If $X$ is ...
4
votes
1answer
36 views

First eigenvalue of Laplacian and Poincaré inequality

Any idea on how to solve: $\int_{\Omega} |\nabla u|^2 d^n x=\lambda_1\int_{\Omega}u^2 d^n x$, with $u\in H^1_0(\Omega)$ and $\Omega\subset\mathbb{R}^n$, and $\lambda_1$ the first eigenvalue of the ...
1
vote
1answer
38 views

$X$ complete normed space $\implies\mathrm B(X,Y)$ complete normed space?

$\newcommand{\N}{\mathbf N}\renewcommand{\leq}{\leqslant}\renewcommand{\geq}{\geqslant} \newcommand{\eps}{\varepsilon}$I was looking through the functional analysis notes of TWK (on his webpage ...
1
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1answer
40 views

Gelfand transform on disk algebra

I tried to prove that if $A$ is the disk algebra then the Gelfand transform is the identity map. The statement can be found here in Theorem 4.4 but it is given without proof. Please can someone check ...
0
votes
0answers
26 views

Check continuity and find norms of linear functionals

1) $\ell^1\ni(x_n)_{n=0}^{\infty}\rightarrow \sum_{n=0}^{\infty}(2013x_{2n}-2012x_{2n+1})\in \mathbb{R}$ 2) $C[0,1]\ni f \rightarrow (a_n f(a_n))_{n=0}^{\infty} \in \ell^\infty$ where ...
0
votes
1answer
19 views

Check continuity of linear functionals

I have to check continuity of these functionals: 1) $\mathbb{R}[X]\ni p \rightarrow X p' \in \mathbb{R}[X]$ with norm $||p||_1=\int_0^1|p(t)|dt$ 2) $C[-1,1]\ni f(t) \rightarrow ...
0
votes
1answer
35 views

Convergence of series of continuous bounded functions

Let $\mathcal{C}$ be the set of continuous bounded mappings from $\mathbb{R}^n$ to $\mathbb{R}^m$. Let $F: \mathcal{C} \rightarrow \mathcal{C}$ be a continuous (with respect to the $\sup$ norm) ...
0
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1answer
37 views

Problem determining eigenvalues of a Hermitian matrix

Suppose that you've got an $n \times n$ irreducible matrix $A$ with strictly positive real entries and eigenvalues $\lambda_i$, $i=1,...,m$, arranged so that $|\lambda_1| > \cdots > ...
0
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1answer
17 views

C_c(R^n,R^m) dense in L^1(R^n,R^m)

I know that $C_c(\mathbb{R}^n)$ is dense in $L^1(\mathbb{R}^n)$. Is the same true for functions from $\mathbb{R}^n$ to $\mathbb{R}^m$? That is is $C_c(\mathbb{R}^n,\mathbb{R}^m)$ dense in ...
0
votes
0answers
14 views

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

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 ...
1
vote
1answer
52 views

Why is $I+T$ invertible for this rank-one operator $T$?

I am working with the following lemma given in the book Topics in Banach Spaces Theory: Let ${(x_n)}_{n=1}^{\infty}$ be a basic sequence in Banach Space $X$. Suppose that there exists a linear ...
2
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0answers
32 views

uniform equivalence to unit vector basis of $\ell_p$

Let $(e_n)$ be the unit vector basis of $\ell_p$, $1\leq p<\infty$. It is well-known that if $(x_n)\subset\ell_p$ is seminormalized and weakly null then it contains a subsequence equivalent to ...
1
vote
1answer
26 views

Is the set $\{ u \in H^1(\Omega) : 0 < a \leq u(x) \leq b\quad \text{a.e.}\}$ closed?

Is the set $\{ u \in H^1(\Omega) : 0 < a \leq u(x) \leq b\quad \text{a.e.}\}$ closed? $\Omega \subset \mathbb{R}^1$ is an interval. There is an embedding into $C^0(\Omega)$. But not sure if this ...
1
vote
1answer
32 views

finding a complex- valued measure

1- Put $A= \Bbb D^{-}\cap {\Bbb D^{c}}^{-}$(Boundery of $\Bbb D$). Let $P= \{p|A; p=$ an analytic polynomial$\}$ and iconsider $P$ as a manifold in $C(A)$. Show that if $\mu$ is a real- valued measure ...
1
vote
1answer
25 views

Eigenvalues of adjoint for residual spectrum.

Statement: Let $T$ be a bounded operator in a Hilbert space $\mathscr{H}$ Show that if $T-\lambda I$ is not dense in $\mathscr(H)$, then $\overline{\lambda}$ is an eigenvalue of $T^*$. Attempted ...
3
votes
1answer
40 views

Function in $H^1$, but not continuous

By the Sobolev embedding theorem, if $\Omega$ is bounded, $H^s(\Omega)\subset C(\Omega)$ for $s>1$, in $\mathbb R^2$. Where I can find a counterexample (if one exists) for the case $s=1$? I mean a ...
0
votes
1answer
47 views

Orthogonal Decomposition => Orthogonal Complements

Hi there can someone prove or disprove the hypothesis: $$\left(X=U\underline{\oplus}V\right)\Rightarrow\left(U^\bot=V,V^\bot=U\right)$$ (I don't require the space to be complete though)