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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11 views

The difference between semicontinuity and hemicontinuity.

For a point-to-set function F, is "upper hemicontinuous" the same as "upper semicontinuous"? If not, then what's the difference?
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1answer
18 views

Vector space clarification

I'm asked to decide if the following are vector spaces. A=$\{f:[0,1] \to \mathbb{R}:\int_0^1|f(x)|dx=0$ $\}$ B= $\{f:[0,1] \to \mathbb{R}:f'(x)+4f(x)=0$ and $f(0)=1 $} C=$\{f:[0,1] \to ...
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1answer
8 views

What is the metric on ${(L^{p}(\Omega))}^N$? (prove that sobolev spaces uniformly convex)

Here what I've done to prove that Sobolev Spaces $W^{(m,p)}(\Omega)$ are uniformly convex for $1<p<\infty$ Given integers $n\geq 1,k\geq0$, we define $N(n,m)$ as the number of multi-indices ...
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0answers
22 views

Non unit version of Stone-Weierstrass theorem

If we assume the Stone-Weierstrass theorem, how to prove the following statement: If $X$ is compact Hausdorff, $ C(X \to \mathbb R)$ is the set of continuous real valued functions. If $ A \subset ...
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20 views

Operator norm of the inverse

If I made no mistake, one can calculate the operator norm of the inverse of any given (invertible) operator $A: V\rightarrow V$ via: \begin{align}\|A^{-1}\| & = ...
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1answer
23 views

Continuity of function between Banach space

The question is: Let B be a Banach space and let f from B into B be a linear map such that f^2=f and both Im(f) and Ker(f) are closed. We want to show that f is continuous. So since Ker(f) is closed ...
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0answers
19 views

Existence of Non-Trivial, Convex, Open Set in $C_{\mathbb{C}}[0,1]$ Under $L^{0}$ Metric

I've been struggling with this problem for the last four hours. The problem is to show that the space of $\mathbb{C}$-valued continuous functions on $[0,1]$ under the metric ...
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1answer
29 views

complex inner product from the real

Let $V$ be a real inner product space. If $W=V\times V$ with the operations $(u_1,v_1)+(u_2,v_2)=(u_1+u_2,v_1+v_2)$ and $(\alpha +i\beta)(u,v)=(\alpha u-\beta v,\alpha v+\beta u)$, where $u, ...
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2answers
22 views

Bijective Bounded Operator Extension: Where do the new elements go to?

Given a dense, proper subset of complete spaces: $$X,Y\text{ both complete and }A\subsetneq\overline{A}=X$$ and an operator between them: $$T:A\to Y\text{ continuous, linear and bijective}$$ Now, ...
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1answer
41 views

Vector spaces isomorphic, then dual spaces isomorphic

If we know that there is a (topological) isomorphism between two Banach spaces $X,Y$ called $\phi \in L(X,Y)$. Then the appropriate isomorphism between the dual spaces $X',Y'$ is given by $\phi' \in ...
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1answer
25 views

Mollification of a function continuous on $\mathbb{R}\backslash\{0\}$, need uniform convergence

We know that if $f:\mathbb{R} \to \mathbb{R}$ is continuous, then its mollification $f_\epsilon$ converges uniformly to $f$ on compact subsets of $\mathbb{R}$ as $\epsilon \to 0$. My question is, ...
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0answers
29 views

Second Derivative of Eigenvalue

Consider the potential $V_t=tx$ and the corresponding Schrödinger operator $H_t=- \frac{\partial ^2}{\partial x^2}+V_t$ on $L^2([0,R])$ with Neumann or Dirichlet boundary conditions. Let ...
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1answer
17 views

Isometry <=> Adjoint left inverse

Is it true that: $$T\text{ isometric}\iff T^*\text{ left inverse to }T$$ Obviously: $$\text{"}\Rightarrow\text{": }\langle x,\tilde{x}\rangle=\langle Tx,T\tilde{x}\rangle=\langle ...
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11 views

Isometry: Adjoint = Leftinverse

Given an isometric operator is it true that its adjoint is necessarily leftinverse? My attempt goes like this: $$\langle x,\mathbb{1}\tilde{x}\rangle=\langle x,\tilde{x}\rangle=\langle ...
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0answers
16 views

An easy (I guess) exercise about limit inferior

Here is an exercise: Let $\{a_{n}\}$ be a positive increasing sequence, can we prove that: lim inf$_{n\rightarrow\infty}\frac{a_{n}}{a_{n-k}}\leq$ lim inf$_{n\rightarrow\infty}a_{n}^{k/n}$? Could ...
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0answers
12 views

A question about subrepresentation in C*-algebra

Recall A function $\phi: \Gamma\rightarrow\mathbb{C}$ is said to be positive definite if the matrix $$[\phi(s^{-1}t)]_{s,t\in F}\in M_{F}(\mathbb{C})$$ is positive for every finite set $F\subset ...
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1answer
20 views

Is the dual of a complete topological vector space always complete?

Let $X$ be a complete topological vector space (over $\mathbb{C}$ say), and $X'$ its dual with the weak*-topology. Then is $X'$ always complete? You may assume $X$ is locally convex if you like.
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13 views

What is the convergence criterion for linear fixed-point iteration in Banach space?

Consider an iterative process of the form $x^{n+1}=A x^n + b$. When $A$ is a linear operator in $\mathbb R^n$ then the criterion of convergence is $\rho(A)<1$, where $\rho(A)$ is spectral radius of ...
2
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0answers
22 views

Chain rule for weak derivatives of $f(u)$ where $f'$ is not bounded but $u$ is?

Let $f:\mathbb{R} \to \mathbb{R}$ be $C^1$. Suppose $u$ has a weak derivative $u_x$. I want the chain rule $$\partial_x (f(u)) = f'(u)u_x$$ to hold. We know this holds if $f'$ is bounded. But I don't ...
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0answers
26 views

Riesz Representation Theorem

I am unfamiliar with Quantum Mechanics and all that stuff. I have recently studied Riesz Representation Theorem , I got to know that it justifies ket and the bra notation. Can anyone please give an ...
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0answers
18 views

Is the function space of a Banach space with product topology a Frechet–Urysohn space?

Consider $E^X$ the space of functions $f: X\to E$ where $X$ is a set and $E$ is a Banach space over $\mathbb{R}$ or $\mathbb{C}$. Using the product topology, then if a sequence of function $f_n \in ...
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1answer
21 views

$f, \hat {f} \in L^{p} \cap L^{\infty} \implies f\in A(\mathbb R)$?

$1\leq p \leq \infty $. We put, $$X_{p}= \{f\in L^{p}(\mathbb R)\cap L^{\infty}(\mathbb R) :\hat{f}\in L^{p}(\mathbb R)\cap L^{\infty}(\mathbb R)\};$$ and we consider the algebra of Fourier ...
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3answers
69 views

a continuous function, satisfying $f(α) = f(β) +f(α −β)$ for any $α, β ∈ \mathbb{R}$ [duplicate]

Hi need some help with this problem: Assume $f : \mathbb{R} → \mathbb{R}$ is a continuous function, satisfying $f(α) = f(β) +f(α −β)$ for any $α, β ∈ \mathbb{R}$, and $f(0) = 0$. Then $f(α) = α ...
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0answers
15 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 ...
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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
27 views

Linear Projections: Bounded/Continuous?

Are linear (nonorthogonal) projections on (pre) Hilbert spaces necessarily bounded/continuous? (can you give a proof or counterexample)
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1answer
18 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
48 views

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

Is there $f:\ell^1\to \ell^\infty$ so that $f$ is surjective $\forall x,y\in \ell ^1, \|x-y\|_1=\|f(x)-f(y)\|_\infty$
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0answers
21 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 $a_n x_n$ ...
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1answer
29 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
15 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
29 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
34 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 ...
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1answer
16 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
13 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), ...
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0answers
31 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
20 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 ...
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1answer
18 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 $f(\bar{x}) = \bar{x}$). ...
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2answers
67 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, ...
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0answers
21 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 ...
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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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1answer
19 views

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

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 ...
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1answer
37 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 ...
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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".
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1answer
27 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
30 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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1answer
26 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 $A,B$ ...
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1answer
20 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 ...
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vote
0answers
105 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
29 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 ...