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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16 views

Question about Hahn-Banach theorem

Let $(X,\|\cdot\|_1)$ and $(Y,\|\cdot\|_2)$ be normed spaces, and $X\subset Y$. If each $f\in (X,\|\cdot\|_1)^\ast$ extends to a bounded linear functional in $(Y,\|\cdot\|_2)^\ast$ with same norm, ...
0
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0answers
8 views

Equicontinuous and pointwise bounded implies compact

Please can you check my proof? I proved: Let $X$ be a compact Hausdorff space and let $C(X)$ denote the space of continuous functions $f: X \to \mathbb R$ endowed with the sup norm $\|\cdot ...
-2
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2answers
21 views

How many functions are there to A set with A elements from A set with B elements?

How many functions are there to a set with A elements from a set with B elements? I'm looking for a short answer for this question. Please help me
1
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0answers
19 views

Is the $C^r(M, N)$ space, with the strong (Whitney) topology, a Fréchet-Urysohn space?

Given smooth, non-compact manifolds $M$ and $N$, consider the function space $C^r(M, N)$. Equipped with the strong (Whitney) topology, this space is Hausdorff and Baire. It is, however, not first ...
2
votes
2answers
45 views

Is this linear functional bounded? Find the norm.

$$\ell^2\ni (x_n)\rightarrow2x_{1}+28x_2+35 x_{3}$$ I think it can be bounded: $$|2x_{1}+28x_2+35 x_{3}| \le |2x_{1}|+|28x_2|+|35 x_{3}| \le 65 (\sum_{n=0}^{\infty}|x_n|^2)^{1/2}$$ But I can't find ...
1
vote
1answer
22 views

Complex exponential is not Fourier multiplier on $L^p$

I am having difficulty to show that the function $m(\xi):= e^{i|\xi|^2}$ is not a Fourier multiplier on $L^p$ when $p\neq 2$. Note that $m:\mathbb{R}^n\to \mathbb{C}$ is called an $L^p$ Fourier ...
-1
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0answers
8 views

side information from equations

Consider system of equations (linear or nonlinear) generated from financial or physical problems. First, the solution to the equations will give information about the problem. Besides the solution, is ...
-5
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0answers
33 views

A separable Hilbert space [on hold]

A separable Hilbert space.prove the followong problem:
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0answers
17 views

Set of limit points of Riemann Integrable functions

I've looked around for answers to this question. It seems like perhaps I don't have enough knowledge of functional analysis to figure out the answer (or even understand the answer), but I'm intrigued. ...
-4
votes
0answers
17 views

let E=C[X] be a normed space and T∈ L(E)… prove that.. [on hold]

Let E=C[X] be a normed space and T∈ L(E). And let $$\||P||_\ = \left\{ \sum ||P^{(n)}||_\infty, \; \; 0 \leq n \leq ∞ \right\}.$$ where $\||P||_\infty$=sup|p(x)|, 0≤x≤1 1- Justify that T:E→E ...
1
vote
0answers
19 views

Compact and bounded if and only if $X$ is finite dimensional

I tried to prove the theorem Let $X$ be a Banach space. Then $K(X) = B(X)$ if and only if $X$ is finite-dimensional. Please could someone check my proof? Let us use the fact that a linear ...
1
vote
1answer
14 views

Show compactness of an operator with Arzelà–Ascoli

We have $K\colon L^{2}(a,b) \rightarrow L^{2}(a,b)$ such that $ Kf(t)=\sum_{j=1}^{n}\phi_{j}(t) \int_{a}^{b} \psi_{j}(S) f(s)ds$ where $\phi_{j} ,\psi_{j} \in L^{2}(a,b)$. We want to show that K is ...
-2
votes
0answers
35 views

Let $\,E = C([0, 1])$. Set $\,X =(E,||.||_\infty)$ and $\,Y =(E,||.||_1)$. … [on hold]

Let $E = C([0, 1])$. Set $\,X =(E,||.||_\infty)$ and $\,Y =(E,||.||_1)$. Let us consider the identity $I :X→Y$. Prove that I is continuous and bijective. Calculate $\,||I||$. Prove that $I^{-1}$ is ...
0
votes
0answers
30 views

Prove that B is bounded ?? [on hold]

Let G be a Banach space and let B be a subset of G. Suppose that f∈G* we have f(B) = {f(x); x∈B} is bounded in R. Prove that B is bounded. Such that G* is the dual space.
1
vote
0answers
31 views

Compact operators is a linear subspace of bounded operators

Let $X,Y$ be Banach spaces. Let $B(X,Y)$ be the set of bounded linear operators and let $K(X,Y)$ be the set of compact linear operators. I want to prove that $K(X,Y)$ is a vector subspace of ...
1
vote
0answers
23 views

Fourier transform of $e^{it\sqrt{a^2+x^2}}$

The question is clear, I came up with this Fourier transform to calculate while searching explicit solutions for a PDE, but I don't even know if it is feasible. $ \mathcal{F}_x(e^{it\sqrt{a^2+x^2}}) ...
-1
votes
0answers
15 views

prove that F is dense in C(X×Y,R) ? any help! [on hold]

Let $X,Y$ be compact metric spaces. Let $$F= \left\{ \sum A_i f_i(x) g_i(y), \; f_i \in C(X,\mathbb{R}), \; g_i \in C(Y,\mathbb{R}), \; 1 \leq i \leq n \right\}.$$ Prove that $F$ is dense in $C(X ...
0
votes
1answer
15 views

Nonexpansive Affine Operators in Hilbert spaces

Let $H$ be a Hilbert space with real inner product $\langle \cdot, \cdot \rangle : H \times H \rightarrow \mathbb{R}$. Let $T$ be an affine operator. Show that $T$ is nonexpansive, i.e., $\left\| ...
1
vote
1answer
41 views

Notation question A $\subset \subset B$

I am a bit confused about the notation A $\subset \subset B$ used in functional analysis. The definition I have says: $A \subset \subset B$ iff $A \subseteq B$ and $\bar{A}$ compact in $B$. Wikipedia ...
0
votes
1answer
57 views

Theorem with an example

I have this theorem In the paper they give an example: But here $H_1$ is not satisfied ! How to correct it please? http://mathoverflow.net/questions/163788/theorem-with-an-example
1
vote
1answer
16 views

An unproved statement from Gohberg and Krein

In the book: Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space, by Gohberg and Krein, page 3, there is an unproved statement that seems to be pulled from nowhere. Let P ...
3
votes
0answers
38 views

Trouble finishing a (direct) proof that $\ell^2(A)$ is a complete metric space

Let $A$ be any non-empty set. We can define summations of non-negative numbers over this index set by using a supremum of summations over finite subsets of $A$. That is, $$\sum\limits_{\alpha \in A} ...
-5
votes
0answers
36 views

need help to prove the following problem [on hold]

Show that $s_n$ is a Cauchy sequence.
1
vote
0answers
26 views

Invertible iff Bounded below and dense range

Statement: Given a Hilbert space $\mathscr{H}$ and $\mathscr{K}$ and a bounded operator $A \in \mathscr{B}(\mathscr{H}, \mathscr{K})$. Show that $A$ is invertible if and only if $A$ is bounded below ...
1
vote
1answer
43 views

Norm of linear functional with e

I have to find norm of this functional: $$\ell^{4/3} \ni (x_n)_1^{\infty} \rightarrow \left(\left(1+\frac{1}{n}\right)^n x_n\right)_1^{\infty} \in \ell^{4/3}$$ I proved that this functional is ...
-2
votes
0answers
26 views

i need to show this problem.(functional analysis) [duplicate]

Show that: The annihilator of a non empty set M in an inner product space X , is a closed subspace of X.
1
vote
1answer
29 views

Tanh function representation for conditional function

I have a condtion as $$T(x)= \begin{cases} -1 & \text{if }x <a \\ 0 & \text{if }a\le x \le b \\ 1 & \text{if }x >b \end{cases} $$ I want to approximate the above condition as ...
1
vote
1answer
81 views

i need help to prove this problem(functional analysis)

show that the annihilator of a set M in an inner product space X is a closed subspace of X.
1
vote
1answer
19 views

Check continuity of linear functionals and find norms

1) $c_{00} \owns (x_n) \mapsto \sum_{n=0}^{\infty} x_n \in \mathbb{K}$ where $c_{00}$ is a space of sequences that are eventually equal to $0$ with sup norm 2) $\ell^\infty \owns (x_n) \mapsto ...
-1
votes
1answer
41 views

equi integrablity

enter link description hereLet $\Omega$ be an open subset of $\mathbb{R}^n$, and let $(f_n)$ be a sequence of measurable functions, $f_n \in L^1(\Omega)$, which is bounded in $L^1(\Omega)$ ($f_n \in ...
0
votes
0answers
20 views

Showing equivalence of weak convergence on closed and open intervals

Quick question. Let $I$ be an open bounded subset of $\mathbb{R}^{n}$. If I am given that $u_{m},u \in W^{1,\infty}(I)$ and I want to show that $u_{m} \rightharpoonup^{*} u$ in $L^{\infty}(I)$. Then I ...
0
votes
0answers
27 views

Is Arzela-Ascoli with equicontiuous or uniformly equicontinuous

I am still working on this proof of Arzela-Ascoli but now I noticed that in my statement of the theorem I used ''equicontinuous'' to mean ''uniformly equicontinuous''. At least in the direction I ...
-1
votes
1answer
44 views

norm of integral operator in $C([0,1])$

If we define on $C([0,1])$ the operator $$ Tf(x) = \int_{0}^{1} K(t,s) f(s) ds$$ where $K$ is a continous function on two variables. I want to show that: $1)$ $||T|| = \displaystyle\max_{t} ...
0
votes
0answers
18 views

Convex Feasibility problem on Hilbert space

Let $H$ be a Hilbert space with real inner product $\langle\cdot, \cdot \rangle: H \times H \rightarrow \mathbb{R}$. Consider a closed convex set $X \subset H$ and let $P_X: H \rightarrow X$ be the ...
0
votes
1answer
29 views

Prove that $F$ is dense in $C(X\times Y,\mathbb{R})$?

Let $X$ and $Y$ be compact metric spaces. Let $$ F= \Bigl\{\sum_{i=1}^n A_i f_i(x) g_i(y): f_i\in C(X,\mathbb{R}),g_i\in C(Y,\mathbb{R}), 1\le i\le n \Bigr\}. $$ Prove that $F$ is dense in $C(X\times ...
1
vote
1answer
29 views

Proof of equivalent characterizations of compact operators

As an exercise I tried to prove the following theorem: If $X,Y$ are Banach spaces and $u \in B(X,Y)$ is a bounded linear operator then the following are equivalent: (1) $u$ is compact ...
0
votes
0answers
17 views

Projection and Pseudocontraction on Hilbert space

Let $H$ be a Hilbert space with real inner product $\langle\cdot, \cdot \rangle: H \times H \rightarrow \mathbb{R}$. Consider a closed convex set $X \subset H$ and let $P_X: H \rightarrow X$ be the ...
1
vote
0answers
23 views

Show: the subspace of compact operators is $\Lambda$-invariant

Let $X$ be a Hilbert space and $A \in \mathcal{L}(X)$ be fixed. Define $\Lambda: \mathcal{L}(X) \rightarrow \mathcal{L}(X)$ by $ \Lambda (T)=A^{*}T+TA,\;T\in \mathcal{L}(X)$. Show: the subspace of ...
0
votes
1answer
28 views

there is a measurable function $f$ on $X$ such that $|{f(x)}|=1$ for a.a $x \in X$ and $\nu(E)=\int_Efd|{\nu}|$ for any $E \in \mathfrak{M}$

any hints on this problem: Let $\nu$ be a finite signed measure on a measure space $(X, \mathfrak{M})$ and let $|{\nu}|$ be its total variation, prove that there is a measurable function $f$ on $X$ ...
0
votes
1answer
51 views

Let $\{f_n\}_{n=1}^\infty$ be non-negative functions and $f_n \to f$ then $f \geq 0$

I have trouble with this question: Let $\{f_n\}_{n=1}^\infty$ be a sequence of non-negative functions in $L^2(0, 1)$, and suppose that $f_n$ converges to a function $f$ in the norm of $L^2(0, 1)$. ...
1
vote
1answer
36 views

The Gaussian Integral

Hi I am trying to calculate the expected value of $$ \mathbb{E}\big[x_i x_j...x_N\big]=\int_{-\infty}^\infty x_ix_jx_k...x_N \exp\bigg({-\sum_{i,j=1}^N\frac{1}{2}x^\top_i A_{ij}x_j}-\sum_{i=1}^Nh_i ...
1
vote
1answer
28 views

Hilbert space inequality $|\langle x,y\rangle | \leq \epsilon ||x||^2 + C_{\epsilon}||y||^2$

In prelim prep I came across 'given $\epsilon$ there exists $C_{\epsilon}$ such that $|\langle x,y\rangle | \leq \epsilon ||x||^2 + C_{\epsilon}||y||^2$. It is asserted without proof, so I've tried ...
3
votes
1answer
24 views

Show that $\lambda \in \sigma(A),$ $\lambda$ not an eigenvalue, implies that $\lambda \in \sigma(A + K)$ where $K$ is compact.

Let $A : H \rightarrow H$ be a bounded linear map where $H$ is a Hilbert space with $\dim H = \infty$. Suppose that $\lambda \in \sigma(A)$ but $\lambda$ is not an eigenvalue. Let $K : H \rightarrow ...
3
votes
0answers
41 views

Subsequences of a basic sequence

Suppose ($x_n$) is a basic sequence in a Banach space $X$, and $Y$ is a closed, infinite co-dimensional subspace of $X$. Can we always find a subsequence ($y_n$) of ($x_n$) such that the intersection ...
0
votes
0answers
38 views

Continuous compactly supported real valued functions on a locally compact and $\sigma$ compact space is separable

I already know that if $X$ is a compact metric space then the space of continuous real valued functions $C(X \to \mathbb R)$ are separable. What I'm trying to prove that if $X$ is a locally compact ...
1
vote
1answer
18 views

Show that M is closed convex and find the minimum norm

Let M={$y=(y_1,...,y_n) \subset C^n: y_1+...+y_n=1$}. Show that M is closed, convex, and find the element of minimum norm in M. Prove M is convex Proof: A set $ M \subset C^n$ is convex if for every ...
0
votes
3answers
62 views

$∫_0^1f(x)x^ndx=0$ for all $n \geq 0$ then $f=0$!

Hi I was thinking about a problem and have a question: we know that if $f∈C([0,1])$ for which $∫_0^1f(x)x^ndx=0$ for all $n \geq 0$ then $f=0$! Now my question is: Do we still have the same when we ...
2
votes
1answer
35 views

sub-algebra of continuous real valued functions without unit must vanish at a point

If $X$ is compact Hausdorff and $A$ a closed subalgebra ( a vector space and closed under multiplication) of $C( X \to \mathbb R)$ the set of continuous real valued functions which separates points, ...
0
votes
1answer
15 views

The orthogonal complement of $M \subset L^2(0,1)$ is the subspace generated by…

The orthogonal complement of $M \subset L^2(0,1)$ is the subspace generated by 1, $e^{2\pi ix}$, and $e^{4\pi ix}$. By definition the orthogonal complement of a subspace $M \subset H$ is the set ${y ...
-3
votes
1answer
35 views

Exercise 3.6: Elementary Functional Analysis By Barbara [on hold]

Let $X=\ell_\Bbb R^\infty$ denote the space of bounded sequences with real entries, in the supremumnorm. Consider the operator $T$ defined on $X$ by $T(x_1, x_2, . . .) = (x_2, x_3, . . .)$; this is ...