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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Characterization of the weak topology

In our functional analysis lecture we defined the weak topology in a what seems to me like a non canonical way, i.e. not as unions of finite intersections of preimages of open sets in the underlying ...
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18 views

A closed set A and compact set B in a topological vector space.

I don't believe this is a repeat question. I have seen it asked before on here, but not in this way. If I take a closed set $A$ in a topological vector space $X$ and a compact set $B$ also in $X$, is ...
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0answers
24 views

What powers of $|x|$ belong to $L^1$?

Prove that $|x|^ {−qp} \in L^{1}(U)$, where $U=B_{1}(0)\subset \mathbb{R}^{n}$. I think I could use polar coordinates to facilitate the work but not sure if it is useful.
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2answers
39 views

Question about vector spaces with the discrete topology

Is it true that every vector space with the discrete topology is a topological vector space? (That is, a topological space with continuous addition and scalar multiplication whose singletons are ...
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26 views

Understanding the set structure of probability theory [on hold]

Since events have their own probabilities and outcomes have their own probabilities. Why don't we just consider only one of events or outcomes directly? What's the motivation to have this set-point ...
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93 views

Real analytic functions

I'm writing because I don't know the usefulness of real analytic functions. I mean, I know that analyticity is something more respect differentiable ($C^\infty$ function), but I don't have in mind a ...
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1answer
18 views

Exercise: Uniform Boundedness Principle and Double dual

Let $X$ be a normed vector space and $(x_{n})$ be a sequence in $X$. Show that if the sequence $f(x_{n})$ is bounded for every $f \in X^{\ast}$, then there exists $C > 0$ such that $\|x_{n}\| < ...
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0answers
26 views

Specific Root of Interpolating Polynomial

We define polynomial $P=(x-\alpha)\cdot g(x)$, where $deg(p)=n-1$, $\alpha \leftarrow \mathbb{Z}_p$. We evaluate $P$ at some $x_i$ values. So we get $(x_1, y_1),...,(x_n, y_n)$, where $P(x_i)=y_i$. ...
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1answer
19 views

Do approximate identities remain approximate identities if one adjoins 1 to a C* Algebra?

If we have a C* Algebra $\mathscr{U}$ without an identity we can adjoin an identity $\mathbb{1}$ in the following way: We take $\mathscr{\tilde U}$ to be the set $\{(\alpha,A); \alpha \in ...
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1answer
20 views

Unitary elements in Banach spaces and subspaces.

Let $F$ be a Banach space and $E$ be a subspace of $F$. Let $e_{0}\in E$ be an element of norm $ 1$ and suppose that span $\{f\in F^{*}:\|f\|=f(e_{0})=1\}=F^{*}$, where $F^{*}$ is the dual space of ...
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1answer
35 views

Is there a name for operators of the type $A: M \to M$

In some theorem in functional analysis I have noticed that it is important to assume that an operator $A: M \to M$ where $M$ is some set plus conditions, as opposed to $A: M \to N, M \neq N$ Is ...
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10 views

Density of intersection of sets with boundary condition

I would like to prove that $$E:=\bigcap_{n\geq 1} \left\{f\in C^2 (\mathbb{R}) :f(0)=\sum_{k\geq 0} f\left(\frac{k}{n}\right)g_n (k)\right\}$$ is a dense subset of: $$F:=\left\{f\in C^2 (\mathbb{R}) : ...
2
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1answer
30 views

Why is the Span of a subset of a linear space defined in such at way?

If I have a subset $M$ of a linear space $E$, we define the linear span of the subset, $M$, as: $$\operatorname{span} M=\bigcap_\alpha \{E_\alpha : E_\alpha \hookrightarrow E\text{ and } M \subseteq ...
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2answers
50 views

intuition of mass function of random variable [on hold]

When we are using $P\{X=x\}$ it seems like intuitively there is a function from $T$ (or measure from $\mathcal{B}(T)$) to $[0,1]$. What is the theoretical foundation behind this intuition?
2
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1answer
35 views

Why does the limit $ \lim_{\varepsilon\to 0+}\int_{\varepsilon}^M \frac{\varphi(x)-\varphi(0)}{x}\ dx$ exist for smooth $\varphi$?

Let $D(\Bbb{R}):=C_c^\infty(\Bbb{R})$ and $$ p.v.(1/x)(\varphi):=\lim_{\varepsilon\to 0}\int_{|x|>\varepsilon}\frac{\varphi(x)}{x}\ dx. $$ for $\varphi\in D(\Bbb{R})$. I'm trying to understand ...
4
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1answer
45 views

Example of Topological Vector Space

Is there a topological vector space such that, for every $x\in X$, there is a proper neighbourhood $V$ of $x$ in $X$ which is convex, but the whole space is not locally convex (i.e. $X$ has a local ...
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1answer
49 views

Example of Heine-Borel Theorem

For a subset $S$ of Euclidean space $\mathbb R^n$, S is closed and bounded if and only if S is compact (that is, every open cover of $S$ has a finite subcover). I need an example or ...
2
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1answer
25 views

Is $\ell_1$ complemented in its double dual $\ell_1^{**}$? (i.e., in $\ell_\infty^*$?)

Quick question, y'all. Is $\ell_1$ complemented in $\ell_1^{**}=\ell_\infty^*$? Yes, I searched Google, and also the standard texts. I can't seem to find an answer, but surely this is known. ...
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0answers
18 views

Show that the continuum of elements $e^{i\lambda t}$ forms a complete orthonormal subset of $B^2$.

Let $X$ be the vector space of all finite linear combinations of functions of the form $e^{i\lambda t}$ ($-\infty<t<+\infty$), where the parameter $\lambda$ is real. An inner product in ...
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1answer
35 views

parallelogramm law and inner products

Is the sum of two norms $||.||_1=\sqrt{(.,.)_1}$ and $||.||_2=\sqrt{(.,.)_2}$, where $(.,.)_1$ and $(.,.)_2$ are the quadratic forms of inner products on a normed linear space, again produced by an ...
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0answers
13 views

“Asymptotic” $\mathbb{R}$-algebras

Definition. By an asymptotic $\mathbb{R}$-algebra, I mean an $\mathbb{R}$-algebra $F$ of functions $\mathbb{R} \rightarrow \mathbb{R}$ satisfying: $$\mathop{\forall}_{f:F}\left[\left(\lim_{x ...
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18 views

What do we call the number that measures how good of an asymptote $g$ is to $f$, and what are the basic results about this number?

Suppose we have a (potentially very complicated) smooth function $f : \mathbb{R} \rightarrow \mathbb{R},$ and we're trying to approximate it (in the limit as the input value goes to $+\infty$) by a ...
2
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2answers
36 views

“Scalar product” of two Lp spaces

I was reading the book A. Lasotta and M. C. Mackey, "Chaos, Fractals, and Noise: Stochastic Aspects of Dynamic", Springer, 1991 On page 27, they defined a ``scalar product'' as follows. Let ...
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1answer
19 views

Equivalence of positivity

Let us have complex matrices and their real decompositions as $H=H_1 + \imath H_2$ and $L = L_1 + i L_2$. Further, $H_1\ge 0$ and $H_2$ is skew symmetric. $L = I - P$ where $P$ is some positive ...
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3answers
42 views

Equivalence of norms problem.

How would I show that $\|\cdot\|_3$ and $\|\cdot\|_\infty$ are equivalent norms on $\mathbb R^2$? I understand that to say two norms are equivalent, then there exist two real constants, $m,M$ such ...
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1answer
47 views

How to prove the uniqueness of probability measure

Probability essentials P-21 Theorem 4.1 (b) Let $(p_\omega)_{\omega \in \Omega}$ be a family of real numbers indexed by the finite or countable set $\Omega$. Then there exists a unique probability ...
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0answers
22 views

Does the “differential” of a unitary representation give continuous operators on the space of smooth vectors?

Let $\pi : G \rightarrow U(H)$ be a strongly continuous unitary representation of a Lie group, $G$, on a Hilbert space, $H$. Let $H_\infty$ be the space of smooth vectors in $H$, those $v$ for which ...
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18 views

Positive-definiteness of a specific function

Is the following function positive-definite $$\varphi(t)=\max\left\{{1-\frac{n-\left|2|t|-n\right|}{2(m+1)},0}\right\}$$ where $0<m<n$ and $0\le |t|<n$. I'm aware of the Bochner's theorem ...
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28 views

Mathematics- fixed point theory [on hold]

If the operator $T$ on a Banach space $X$ is a contraction mapping, then $T$ has a unique fixed point. The inverse of $T$ has also a fixed point but it is never be a contraction mapping-justify.
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1answer
25 views

significance and importance of spectral theorem

I have started recently started Operator Theory and have been introduced to the Spectral Mapping Theorem: If $a \in \mathcal{A}$, where $\mathcal{A}$ is a unital Banach Algebra and $f \in ...
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2answers
52 views

Show that the mapping $f → f~'$ from $C^1([0 , 1])$ to $C([0 , 1])$ is not continuous.

Let $C^1([0 , 1])$ be the subspace of $C([0 , 1])$ consisting of the functions that have a continuous derivative throughout $[0 , 1]$. Show that the mapping $\Psi:f → f~'$ from $C^1([0 , 1])$ to $C([0 ...
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1answer
31 views

Density of spaces $C_0^{\infty}(\mathbb{R})$, $W_2^2(\mathbb{R})$ and $L^2(\mathbb{R})$ in each other

Let's consider following spaces: $L^2(\mathbb{R}) = L^2(\mathbb{R}, \mathbb{C}, \mu_L)$ --- space of $\mathbb{C}$-valued functions defined on $\mathbb{R}$ for which the square of the absolute value ...
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1answer
15 views

Bounded linear functionals in solving PDE

Many theorems in functional analysis, Rietz, Hahn-Banach, for example, are used to find linear functionals in certain spaces. But why are bounded linear functionals useful in solving PDE? This ...
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1answer
26 views

How is Baire category theorem used here?

The following is a doubt that arouse from reading this paper by Bandyopadhyay, Jarosz and Rao. Let $E$ be a Banach space and $E^{*}$ be its dual space. Let $e_{0}$ be an element of norm one in $E$ ...
3
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2answers
51 views

A Banach Space cannot have a denumerable basis:Why is it true?

I came across the following theorem: A Banach Space cannot have a denumerable basis which has been proven in my book. I can't understand why is it true since $\mathbb R$ is a banach space over ...
2
votes
1answer
23 views

About closedness and boundedness of $H=\left\{(x_n)\in \ell^2(\mathbb{N})\mid\sum \frac{x_n}{n}=1\right\}$

Let $H=\left\{(x_n)\in \ell^2(\mathbb{N})\mid\sum \frac{x_n}{n}=1\right\}$. To check which one is true: (a) $H$ is bounded (b) $H$ is closed (c) $H$ is a subspace (d) $H$ has interior points My ...
0
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1answer
26 views

Infinite Hamel basis for Banach spaces

What are some standard examples of Hamel basis for Banach spaces with cardinality >= $\aleph_0$? I tried searching, but couldn't find any.
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0answers
37 views

Where can I find this definition of “expected value”?

I need bibliography or some text about this definition: "Define the expected value of a function by: $E_{t}(x(t))=(\frac{1}{t})\int_{0}^{t} x(s)ds$. " I think that it's statistics or functional ...
0
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1answer
27 views

space of all lipschitz maps is a polish metric space

Suppose that $(X, d_X)$ and $(Y, d_Y )$ are Polish metric spaces. Let $L(X, Y )$ denote the set of all Lipschitz maps from $X$ to $Y$ with the pointwise convergence topology. Show that $L(X, Y )$ is ...
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25 views

Application of Riesz-Markov-Kakutani representation

Let A be finite set, $\{f_i\}_{i=1}^{n}:A\to \mathbb{R}$ non-negative and with the following property for every $\sum \lambda_{i}=1$: for any $g=\sum \lambda_{i}f_i$ there exists $a\in A$ s.t. ...
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2answers
50 views

Show that the following mapping is a contraction.

I have the following problem from a past paper: "Show that the mapping, $$T(x_1,x_2)=\left(\frac{x_1+2x_2}5-1,\frac{x_1-2x_2}7+1\right)$$ is a contraction on $(\mathbb R^2,d_\infty)$." I ...
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31 views

Functional Analysis: Continuity of operators [on hold]

Which of the following operators are continuous: a) $A:L^2 [0,1]\rightarrow L^2 [0,1]$ defined by the formula $\displaystyle (Ax)(t)=\int \limits_{0}^{1} K(t,s) x(s)ds$, where $K(t,s) \in L^2( ...
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1answer
34 views

A closed subspace of a separable Hilbert Space is Separable

Suppose $X$ is a Hilbert Space which is separable. Let $Y$ be a closed Subspace of $X$. I need to show that $Y$ is separable. Since $X$ is separable it has a countable dense subset say $M$. Taking ...
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0answers
26 views

Derivative of mollification

This is in response to a claim made in the second line of the question here, namely: Given the standard mollifier $\eta$ and a locally integrable function $f:U \rightarrow \mathbb{R}^n$, by defining ...
3
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1answer
21 views

Are all normed linear spaces Hausdorff? What about a bounded subset of a normed linear space?

The proof that the dual space of a normed linear space is complete in proposition 5.4 of chapter on Banach spaces (John Conway, functional analysis) consists of restricting the functionals in the dual ...
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18 views

Banach$^*-$ algebra for two different multiplication

Let $B$ be a Banch $^*-$ algebra and we say $f$ is positive linear functional on $B$ if $f(xx^*)\geq 0$ for all $x\in B.$ Let $B$ be a Banach algebra with two different multiplication operations, ...
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26 views

For which $F$ we have $F(f\ast g)= f\ast F(g)$ for all $f,g \in L^{1}$?

Young's inequality tells us that: $L^{1}\ast L^{p} \subset L^{p}, (1\leq p \leq \infty)$ My Question: What are examples of functions $F:L^{1}(\mathbb R)\to L^{p}(\mathbb R)$ with the property ...
2
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0answers
20 views

Orthogonal Complement: Families

Problem Given a Hilbert space $\mathcal{H}$. Consider a family: $$A:\Lambda\to\mathcal{P}(\mathcal{H}):\lambda\mapsto A_\lambda$$ Remind that: $$A\subseteq\mathcal{H}:\quad ...
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0answers
26 views

The convergence of the norm implies the convergence of the sequence via the norm. [duplicate]

I want to prove that: Let $(f_n)$ be a sequence in $L^p$ that converges pointwisely to $f$. Prove that $f_n\rightarrow f$ in $L^p$ iff $||f_n||\rightarrow ||f||$. The "only if" is easy and follows ...
2
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0answers
62 views

Measuring the set-theoretical complexity of sets/spaces encountered in general analysis

In analysis, it is common to encounter subsets of $\mathbb R$ (or even $\mathbb R^n$) which appear to be "well-behaved", especially with regard to properties like being measurable, compactness, etc. ...