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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Questions about the finite-dimensional normed space of polynomials of degree at most two.

Take $X:=P_2([0,1])$, the polynomials of degree at most $2$ over $[0,1]$ and consider the $2$-norm on this space. For any $x\in X$ we have that, $$\|x\|_2=\left(\sum_{i=1}^n|x_i|^2\right)^{1/2}$$ ...
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47 views

Possible ways to induce norm from inner product

Let $ S $ be a normed vector space (over $\mathbb{R}$, say, for simplicity) with norm $ \lVert\cdot\rVert$. Can this norm be induced from inner product only through $\lVert \cdot \rVert = ...
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1answer
28 views

X and Y are normed linear spaces over the same field $F(=\mathbb{C}/\mathbb{R})$, both having the same finite dimension $n$.

X and Y are normed linear spaces over the same field $F(=\mathbb{C}/\mathbb{R})$, both having the same finite dimension $n$. I need to show that $X$ and $Y$ are topologically isomorphic ( A ...
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1answer
34 views

Closed plus finite dimensional in a TVS

If $E$ is a topological vector space (TVS), $F_1$ a closed subspace of $E$, and $F_2$ a finite dimensional subspace of $E$, such that $F_1 \cap F_2=\{0\}$, is $F_1+F_2$ necessarily closed? If yes, are ...
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1answer
46 views

Extending finite rank operators

Suppose $Y$ is a closed subspace of Banach space $X$ and $T:Y\to X$ is a bounded finite rank operator. Can we extend $T$ to $\tilde{T}:X\to X$, in the sense that: $T=\tilde{T}$ on $Y$ ...
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1answer
35 views

How does sketching norms show that they are equivalent?

I have the following statement in my notes: "You might want to check by drawing the sets of all $x\in\mathbb R^2$ such that $\|x\|_1=1$,$\|x\|_2=1$,$\|x\|_\infty=1$ that indeed these norms are ...
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1answer
27 views

Solution to integral equation

Show that the non-linear integral equation $v(x)=\cos^2(x)+\int_0^x e^{-v^2(s)}ds, \ x\in [0,\infty)$ has a solution in $C^1([0,\infty))$. In previous questions of this sort, we have been able to ...
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1answer
67 views

Functions by which one can multiply elements of $L^1_{\text{loc}}$

Let $u$, $\omega\in L^1_{\text{loc}}(\mathbb R^N$) be given. We further assume that $\omega\geq 1$, l.s.c, and satisfies, for a constant $C>0$, $$ \frac{1}{|B(x,r)|}\int_{B(x,r)}\omega(y)dy\leq ...
3
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0answers
23 views

an integral equation in a function with two arguments

Say we are given $C(s,t)=\min(s,t)+\zeta st$. How can we solve $$g(s,t)=C(s,t)+\lambda \int_0^1g(s,u)C(u,t)du.$$ Looking into some text books on integral equations I see that most of the kernels, $C$, ...
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1answer
45 views

Line segment in the unit sphere

I want to prove the following statement Let $X$ be a normed linear space, with linearly independent vectors $x,y$, such that $\|x\|=1=\|y\|$, with $\|x\|+\|y\|=\|x+y\|$, then there is a line ...
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31 views

Can We Always Realize the Value of the Quotient Norm. [duplicate]

Let $(V, \|\cdot\|)$ be a Banach space over $\mathbf R$ and $W$ be a closed subspace of $V$. We know that $V/W$ becomes a normed linear space under the quotient norm $\|\cdot\|_q$ defined as ...
2
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1answer
47 views

Understanding convergence in normed spaces and the language used when talking about norms.

We have the following definition about convergence in a normed space: "Let $(x_n)_{n=1}^\infty$ be a sequence in a normed space $(X,\|\cdot\|)$. We say that $x_n\to x$ in $X$ if, $$d(x_n,x)\equiv ...
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245 views

Examples of infinite dimensional normed vector spaces

In my notes on functional analysis it mentions that $C([0,1]),\ell^p$ and, $\ell^\infty$ are normed vector spaces, and gives some examples of norms that we can define on them. However, it then simply ...
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51 views

Find the iverse of the followning bounded operator?

The following definition and Theorem are given in the book "A short course on operator semigroup" by the author "K-J Engel and R Nagel". Sectoral operator: A closed linear operator $(A,D(A))$ in ...
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1answer
57 views

Infinite series of integrals of $L^2$ functions

I'm hoping someone can help me with this integration problem I've been struggling with. Let $\{f_n\}$ be a sequence in $L^2(\mathbb{R})$ such that $\sum_{n=1}^\infty \lVert f_n\rVert^2_2<\infty$ ...
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1answer
78 views

Showing the compactness of a limit operator.

I was trying to solve this exercise from Kreyszig's book, section 8.1 exercise number 10. My attempt was try to show that the operators in the sequence are bounded, but I don't find it. If this fact ...
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0answers
30 views

Clarification on Wolfram Mathworld's explanation of the connection between Gelfand Transform and Fourier Transform

http://mathworld.wolfram.com/GelfandTransform.html In the definition, what does $x$, $\hat x(\phi)$, and $\phi$ represent exactly if we were to consider definition of the Fourier transform? Can ...
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1answer
37 views

Show $l^p$ embeds in $L^p(0,1)$

Let $l^p$ be the standard sequence space indexed by $\mathbb N$. I've heard it claimed that $l^p$ embeds into $L^p(0,1)$ in such a way that $$L^p(0,1)=l^p\oplus S$$ for some closed subspace $S\subset ...
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1answer
33 views

$C^1(\bar \Omega)$ is a Banach space

My professor gave a proof of the completeness of $(C^1(\bar \Omega),\|\cdot \|_{C^1})$ based on the fundamental theorem of calculus. I though about an alternative and I would like to know whether this ...
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1answer
66 views

Version of Stone Weierstrass for functions not vanishing at infinity

I am trying to see what is known about uniform density of function spaces in $C(\mathbb{R}^n)$ or $C_b(\mathbb{R}^n)$ (bounded continuous functions on $\mathbb{R}^n$). By uniform density, I mean ...
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1answer
28 views

Show for the Hamilton's operator $H$ that $\overline{(H, C_0^{\infty}(\mathbb{R}))} = (H, W_2^2(\mathbb{R}))$ using Fourier transform

Let $V \in C_{b}^{1}(\mathbb{R}, \mathbb{R})$ be a differentiable real-valued function defined on $\mathbb{R}$ bounded with its first derivative. Consider the Hamilton's operator $H$ such that: ...
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1answer
14 views

correspondence between linear functional and function

Any Schwartz or $L^p$ function $g$ can be identified with a linear functional via which way? $T_g(f)=\int gf$ or $T_g(f)=\int g\bar f$ ? I have seen these two different definitions in different ...
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24 views

A question on the Banach Contraction Mapping Principle

The BCMP states that in a complete metric space $X$, a contraction mapping $T$ on $X$ has a unique fixed point, i.e. if $T$ satisfies $d(Tx, Ty) \le k d(x,y)$ such that $0 \le k < 1$, then $T$ has ...
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1answer
17 views

$\lim_{k \to \infty} \langle s_k,e_n \rangle = \langle h,e_n \rangle$

I took a passage from a textbook regarding equivalent conditions of having an orthonormal sequence in a Hilbert space H. Why is the equality $$\lim_{k \to \infty} \langle s_k,e_n \rangle = \langle ...
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2answers
43 views

schauder basis for $\ell_\infty$ [duplicate]

I know that $\ell_\infty$ is not separable, therefore has no Schauder basis. However I cannot understand why the set $\{e_1, e_2, e_3, \dotsc \}$ where $e_1=(1,0,0,\dotsc), e_2=(0,1,0,0,\dotsc), ...
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1answer
47 views

Example of a non-convex set for which A + A = 2A

Give an example of a non-convex subset $A$ of a (real / complex) vector space $V$ for which $$A + A = 2A$$ Here the sum / multiplication with a scalar of a subset is defined in the obvious way. I ...
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30 views

Hahn Banach Theorem: Clarification on meaning of extending a functional?

Hahn Banach Theorem: Given linear (vector) space $\mathbb{X}$, define $u \in \mathbb{L} \subset \mathbb{X}$, $A,B,C$ functionals, A sublinear. $A:\mathbb{L} \to \mathbb{R}, B:\mathbb{L} \to ...
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1answer
28 views

Partial Isometries: Final

Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider an operator: $$J\in\mathcal{B}(\mathcal{H},\mathcal{K}):\quad P:=J^*J$$ By the C*-property: $$J=JJ^*J\iff P^2=P=P^*$$ Note that in any ...
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0answers
41 views

Construct an operator that fixes the equivalence class of Cauchy sequences

Let $X$ be a Banach space and $\overline{X}$ be its unique completion. We know that $\overline{X}$ can be partitioned into equivalence classes of Cauchy sequences via the relation $\sim$: $$ \{x_n\} ...
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0answers
14 views

An analytical expression for the degree of a map from the sphere to itself

Good morning to everyone. I have found in this paper the following statement (not verbatim): "Let $\phi$ be a smooth map from $\mathbb{R}^2$ to the $3$-dimensional sphere $S^2$ which is constant far ...
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1answer
48 views

A nice Application of Baire category Theorem.

Let $f_n$ be continuous functions on a complete metric space $X$ such that $f_n(x)> 0$ for every $x \in X$. Let $A ={x \in X \mid \liminf f_n(x) =0 }$. Prove that $A$ is a countable intersection ...
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0answers
95 views

Is “almost all function” a well defined concept?

I am working on a problem which has well defined properties for the vast majority of all PDFs. I would like to make a quantitative statement along the lines of "for almost all distributions, P holds". ...
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1answer
76 views

What is the difference between an function and functional?

Can someone give an example that would point out the difference between a function and a functional in a very simple way?
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64 views

Does anybody know the definition of $C^{2+\alpha, 1+\frac{\alpha}{2}}(\Omega)$, where $0<\alpha<1$?

I hope someone can give me the definition of the following: $C^{2+\alpha, 1+\frac{\alpha}{2}}(\Omega)$ for some $0<\alpha<1$ and some domain $\Omega$. In this context they also talk about ...
2
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1answer
25 views

(Operator) norm inequality for continuous functions

Let $f,g$ be two non-negative continuous functions on $[0,\infty)$ satisfying $f(t)g(t)=t,$ $\forall t\in[0,\infty)$. Let be $A$ be a bounded linear operator acting on a Hilbert space. Then I was ...
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0answers
20 views

Does $\chi_{A_\epsilon} \to \chi_{\{x : f(x) = 0\}}$ if $A_\epsilon = \{ x \in \Omega : 0 \leq f(x) < \epsilon\}$?

Define $A_\epsilon = \{ x \in \Omega : 0 \leq f(x) < \epsilon\}$ where $f$ is a given function say in $L^1(\Omega)$. Is it true that $$\chi_{A_\epsilon} \to \chi_{\{x : f(x) = 0\}}$$ pointwise? ...
2
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0answers
23 views

Variation problem : Euler equation using direct method

Consider a map $I : C^1[0,1]\rightarrow \mathbb{R}$ defined by $$I (f)=\int_0^1\frac{1}{2}(f'(x))^2 -V(f(x))dx$$ where $V(t)\leq 0$ for all $t \in \mathbb{R}$ is smooth. Let $\Gamma=\{f\in C^1[0,1]|\| ...
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1answer
12 views

If $u_n \rightharpoonup^* u$ in $L^\infty$, does $\int u_n^+ \to \int u^+$?

Let $\Omega$ be bounded. Suppose that $u_n \rightharpoonup^* u$ in $L^\infty(\Omega)$ and $u_n \to u$ in $H^{-1/2}(\Omega)$ (that is negative a half, not a typo). Does this somehow imply that ...
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2answers
69 views

Are all matrices linear operators?

Given $A \in \mathbb{K}^{n\times m}$ a matrix, can we think of $A$ as an operator? In what context do matrices satisfy the definition of operator?
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1answer
26 views

a nontrivial inequality in the proof of weak solution of biharmonic equation

Hi I am looking at the post discussed about weak solution of biharmonic equation Proving unique weak solution. I am having trouble verifying statement 2: The bilinear operator is coercive, The claim ...
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0answers
47 views

Quotient spaces - $\Bbb R^1\hookrightarrow \Bbb R^3$

I am trying to understand quotient spaces, and I constructed my own example to do this: $(\Bbb R=\{(a,0,0)|a\in \Bbb R^1\}) \hookrightarrow (\Bbb R^3=\{(\alpha,\beta,\gamma)|\alpha,\beta,\gamma \in ...
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1answer
78 views

Intuitive functional analysis book

I want to know a functional analysis book like Terence tao's real analysis and measure theory book, full of intuition. I am aware of linear algebra, real analysis, measure theory, Probability theory.
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1answer
55 views

Clarification on the difference between Brouwer Fixed Point Theorem and Schauder Fixed point theorem

From Zeidler's Applied Functional Analysis Brouwer The continuous operator $A:M \to M$ has a fixed point provided $M$ is compact, convex, nonempty set in a finite dimensional normed space ...
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48 views

Is functional analysis related to or used in algebraic geometry in any way?

I'm curious about whether there's a link (and, no, this question was not motivated by the fact that Grothendieck used to be a functional analyst!) between these two subjects. Are the techniques from ...
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1answer
41 views

Need help proving $n(T)=n(T^*)$ for finite dimensions.

In my book this is showed: Let H and K be complex Hilbert spaces and let $T\in B(H,K)$. There exists a unique operator $T^* \in B(K,H)$ such that $(Tx,y)=(x,T^*y)$ for all $x\in H$ ...
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1answer
52 views

A very simple question: what spaces of function does the laplace transform map from and into?

Given a function $f$, we can write $f:\mathbb{R} \to \mathbb{R}$ to denote that $f$ takes a number from $\mathbb{R}$ into $\mathbb{R}$. Easy enough. Given the laplace transform operator ...
2
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1answer
73 views

Sobolev spaces over closed domains.

I am currently working through books on Sobolev spaces and I notice that these spaces are almost always defined over open domains, i.e. we look at $W^{m,p}(\Omega)$, where $\Omega$ is open. Because ...
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2answers
31 views

Show that function is in L^2

I'm going through a paper and I came across the following statement: Given $\mathbf{q}_h \in \mathbf{V}_h(\Omega)$ we have to show that $\nabla\cdot\mathbf{q}_h$ is well defined and in ...
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1answer
78 views

Is there a $C^{*}$ algebra with these properties

Is there a unital C* algebra A which is NOT simple but satisfies the following two conditions: 1)A has trivial center 2)A has a faithful trace such that every zero trace element lies in the closure ...
3
votes
1answer
69 views

concept of the classification of $C^\ast$-algebras, introduction/overview

I don't have a specific mathematical problem at the moment but nevertheless I hope, my question is suitable for math.stackexchange. I'm interested in $C^\ast$-algebras and I would like to begin with ...