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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Eigenvalues and eigenfunctions for the Fredholm integral operator $K(g) = \int_0^1 e^{x t} g(t) dt$.

I would like to compute the eigenvalues and eigenfunctions for the Fredholm integral operator $$K(g) = \int_0^1 e^{xt} g(t) dt.$$ The sources I've checked* seem to say that the process is fairly ...
11
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8answers
822 views

For $f$ continuous, show $\lim_{n\to\infty} n\int_0^1 f(x)x^n\,dx = f(1).$

Suppose $f:[0,1]\to \mathbb{R}$ is continuous. Show that  $$\lim_{n\to\infty} n\int_0^1 f(x)x^n\,dx = f(1).$$ My answer so far: First I want to assume that $f\in C^1$. Then  $$n\int_0^1f(x)x^n\,dx =...
11
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4answers
938 views

Is there a continuous bijection between an interval and a square: $[0,1] \mapsto [0,1] \times [0,1]$?

Is there a continuous bijection from $[0,1]$ onto $[0,1] \times [0,1]$? That is with $I=[0,1]$ and $S=[0,1] \times [0,1]$, is there a continuous bijection $$ f: I \to S? $$ I know there is a ...
11
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4answers
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Why is a function space considered to be a “vector” space when its elements are not vectors?

I am confused by the notion of a function space. For example consider the basis $\{1, x, x^2\}$ which is the basis for the vector space of all polynomials of degree at most $2$. What is the notion of ...
11
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3answers
4k views

A question about Banach reflexive space

How to show that a Banach space $X$ is reflexive if its dual $X'$ is reflexive?
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2answers
2k views

$L^1$ and $L^{\infty}$ are not reflexive

I want some proof for the following statement : $L^1$ and $L^{\infty}$ are not reflexive. Can anyone help me, please? or reference me?
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1answer
9k views

The $ l^{\infty} $-norm is equal to the limit of the $ l^{p} $-norms. [duplicate]

If we are in a sequence space, then the $ l^{p} $-norm of the sequence $ \mathbf{x} = (x_{i})_{i \in \mathbb{N}} $ is $ \displaystyle \left( \sum_{i=1}^{\infty} |x_{i}|^{p} \right)^{1/p} $. The $ l^{\...
11
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3answers
1k views

What is the insight behind the Lebesgue integral?

Edit 3: OK, I had an insight, inspired in part by Ben-Blum Smith's comment, and the post he linked to. (I have no idea if this insight is right; it's barely a hunch, and that's why I'm not submitting ...
11
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4answers
549 views

Why isn't every element of the spectrum an eigenvalue? (Where is the error in my proof?)

My book defines the spectrum like this: Let $H$ be a complex Hilbert space, let $I \in B(H)$ be the identity operator and let $T \in B(H)$. The spectrum of $T$, denoted $\sigma(T)$, is defined ...
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2answers
236 views

Can you equip every vector space with a Hilbert space structure?

Suppose that we have a vector space $X$ over the field $\mathbb F \in \{ \mathbb R, \mathbb C \}$. Question: Does there exist a Hilbert space $\widehat X$ over $\mathbb F$ such that $\widehat X$, ...
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2answers
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Some basics of Sobolev spaces

Let $W^{m,p}(\Omega) = \{ f \in L^p(\Omega): D^\alpha f \in L^p(\Omega) \text{ for multi-indices } |\alpha| \leq m\}$, where $D$ denotes the weak derivative. Let $W_0^{m,p}$ denote the closure of $C_c^...
11
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1answer
268 views

Can limits be thought of as linear functionals (or operators, depending on context)?

Ok so I just started Calc I this summer and since I already feel pretty comfortable with it from high school, I'm trying to gain a more rigorous perspective on it. I already know that limits behave ...
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2answers
506 views

Selfadjoint compact operator with finite trace

I have a compact selfadjoint operator $T$ on a separable Hilbert space. For some fixed orthonormal basis, the operator's diagonal is in $\ell^1(\mathbb{N})$. Can we conclude that $T$ is trace ...
11
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2answers
179 views

Connectedness property of $R^2$

My class teacher proposed this problem which seem very interesting. If we remove countably many open disc from $R^2$. Is the remaining space still be path connected. I have done the problem ...
11
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2answers
692 views

Counterexample for the solvability of $-\Delta u = f$ for $f\in C^2$

I'm learning for an exam and I'm surprised by the following statement that is given without proof or example: Let $\Omega\subset\mathbb{R}^n$ be open, bounded and connected and let $f\in C^0(\...
11
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1answer
2k views

$T$ surjective iff $T^*$ injective in infinite-dimensional Hilbert space?

Let $T:H_1\rightarrow H_2$ be a bounded linear operator where $H_1$ and $H_2$ are Hilbert spaces. The Hilbert-adjoint is defined to the the operator $T^*:H_2\rightarrow H_1$ such that $\langle Tx,y\...
11
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1answer
477 views

Renorming $\mathcal{B}(\mathcal{H})$?

Consider the Banach space of all bounded operators $\mathcal{B}(\mathcal{H})$ on a (separable if you wish) Hilbert space $\mathcal{H}$ with the operator norm. Can we renorm this space to a strictly ...
11
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1answer
3k views

Invertibility of compact operators in infinite-dimensional Banach spaces

Let $X$ be an infinite-dimensional Banach space, and $T$ a compact operator from $X$ to $X$. Why must $0$ then be a spectral value for $T$? I believe this is equivalent to saying that $T$ is not ...
11
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2answers
580 views

Is the closedness of the image of a Fredholm operator implied by the finiteness of the codimension of its image?

Let $X$ and $Y$ be Banach spaces. A bounded operator $T\colon X\to Y$ is called Fredholm iff The dimension of $\ker(T)$ is finite, The codimension of the image $\mathrm{im}(T)$ is finite, The image $...
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2answers
457 views

Non-Banach, completely metrizable normed vector space

Does there exist a normed vector space $(X,\|\cdot\|)$ over $\mathbb R$ or $\mathbb C$ such that the metric induced by the norm $(x,y)\mapsto\|x-y\|$ is not complete; but there exists some other ...
11
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2answers
483 views

Engineer: Unable to appreciate importance of L2 space

Can someone provide me with some material to make clear of the importance of $L^2$ space in engineering/physics? Having a background in all the introductory mathematics courses offered in ...
11
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1answer
818 views

About example of two function which convolution is discontinuous on the “big” set of points

I want to ask about example of real valued functions defined on the real line such that their convolution exist in every point and is discontinuous on a "large" set, for example on each point of some ...
11
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1answer
869 views

Why no trace operator in $L^2$?

We have trace operator which allows us to define boundary values of an $H^1$ function. This is because of the fact that $C^\infty$ is dense in $H^1$ under the $H^1$ norm, I believe. I'm sure either $...
11
votes
3answers
1k views

Applications of the Hahn-Banach Theorems

Question: What are some interesting or useful applications of the Hahn-Banach theorem(s)? Motivation: Most of the time, I dislike most of Analysis. During a final examination, a question sparked my ...
11
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1answer
1k views

Metric and Topological structures induced by a norm

While proving that some normed spaces were complete, two questions came to my mind. They relate the topological and the metric structures induced by a norm. 1) Is it possible to find two equivalent ...
11
votes
2answers
222 views

General “Hodge theorem”

I know basically zero Hodge theory, so this question might be weird. Let $$A \stackrel{S}{\longrightarrow} B \stackrel{T}{\longrightarrow} C$$ be a sequence of closed, densely defined maps of ...
11
votes
3answers
2k views

What's the connection between the Laplace transform and the Fourier transform?

Both the Laplace transform and the Fourier transform in some sense decode the "spectrum" of a function. The Laplace transform gives a power-series decomposition whereas the Fourier transform gives a ...
11
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1answer
2k views

Norm on a Hölder's space

I want to prove that Hölder space is a Banach space under the "Hölder Norm" ie. $\|\cdot\|_{C^{k,\alpha}}$. Any hints would be appreciable .
11
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2answers
147 views

What is the main purpose of learning about different spaces, like Hilbert, Banach, etc?

I just started to learn about functional analysis and have started to learn about various spaces, like $L^{p}$, Banach, and Hilbert spaces. However, right now my understanding is rather mechanical. ...
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3answers
567 views

A theorem due to Gelfand and Kolmogorov

For any topological space $X$, we can define $C(X)$ to be the commutative ring of continuous functions $f\,:\,X\rightarrow \mathbb{R}$ under pointwise addition and multiplication. Then $C(-)$ becomes ...
11
votes
2answers
992 views

A net version of dominated convergence?

Let $G$ be a locally compact Hausdorff Abelian topological group. Let $\mu$ be a Haar measure on $G$, i.e. a regular translation invariant measure. Let $f$ be fixed in $\mathcal{L}^1(G, \mu)$. ...
11
votes
1answer
150 views

When is an operator on $\ell_1$ the dual of an operator on $c_0$?

Suppose $T:\ell_1\to\ell_1$ is a continuous linear operator. When can we say that $T$ is a dual, or adjoint, of an operator on $c_0$? In other words, under what conditions can we find a continuous ...
11
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1answer
348 views

$(n^2 \alpha \bmod 1)$ is equidistributed in $\mathbb{T}^2$ if $\alpha \in \mathbb{R} \setminus \mathbb{Q}$

I did the following homework question, can you tell me if I have it right? We want to show that the sequence $(n^2 \alpha \bmod 1)$ is equidistributed if $\alpha \in \mathbb{R} \setminus \mathbb{Q}$. ...
11
votes
1answer
757 views

Does separability follow from weak-* sequential separability of dual space?

Let $E$ be a Banach space. Suppose that $E'$ is weakly-* sequentially separable, that is, that there exists a countable $D \subset E'$ s.t. every $x' \in E'$ is a limit point of a ...
11
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2answers
145 views

Properties of function $f(x) = (1 + x^2)^{-\alpha/2}(\log(2+x^2))^{-1},\text{ }x \in \mathbb{R}$ with $0 < \alpha < 1$.

Consider the function$$f(x) = (1 + x^2)^{-\alpha/2}(\log(2+x^2))^{-1},\text{ }x \in \mathbb{R},$$with $0 < \alpha < 1$. How do I see that $f \in W^{1, p}(\mathbb{R})$ for all $p \in [1/\alpha, \...
11
votes
1answer
246 views

Is the function characterized by $f(\alpha x+(1-\alpha) y) \le f^{\alpha}(\alpha x)f^{1-\alpha}(y)$ convex?

Is a non-negative function $f(x)$ convex ? If for $x \ge y$ it satisfies for any $\alpha \in [0,1]$. \begin{align} f(\alpha x+(1-\alpha) y) \le f^{\alpha}(\alpha x)f^{1-\alpha}(y) \ \text{ eq....
11
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2answers
329 views

Algebraically flavoured functional analysis book

I'm looking for a book on functional analysis that would suit someone who is more algebraically/geometrically oriented and seeks to learn the subject with the goal of using it later for geometric ...
11
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1answer
135 views

There is no continuous mapping from $L^1([0,1])$ onto $L^\infty([0,1])$

There is no continuous mapping from $L^1([0,1])$ onto $L^\infty([0,1])$. Proof: suppose $T:L^1 \rightarrow L^\infty$ continuous and onto. $L^1$ is separable, let $\{f_n\}$ be a countable dense ...
11
votes
3answers
173 views

Not every function on $[0,1]$ is a pointwise limit of continuous functions on $[0,1]$

How to show that not every $\mathbb{R}$-valued function on $[0,1]$ is a pointwise limit of continuous $\mathbb{R}$-valued functions on $[0,1]$? There is a theorem that states that the set of points ...
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2answers
620 views

Extension and trace operators for Sobolev spaces

Given that $\Omega \subset \mathbb{R}^{n}$ is an open, convex, Lipschitz bounded set. Let $O \subset \Omega$ be open bounded set then consider. $$u_{m} \rightharpoonup^{*} u \text{ in } W^{1,\infty}_{...
11
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1answer
172 views

Non-divergence form of a 2nd order PDE

This might be a trivial question but I'm very rusty with regards to calculus and am new to PDEs. How would you write the following second order quasilinear equation in it's non-divergence form: The ...
11
votes
3answers
366 views

Problem with infinite product using iterating of a function: $ \exp(x) = x \cdot f^{\circ 1}(x)\cdot f^{\circ 2}(x) \cdot \ldots $

[update]: I made the question more precise, more general and added a follow up question Considering the iteration of functions (with focus on the iterated exponentiation) I'm looking, ...
11
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1answer
654 views

Motivation of Feynman-Kac formula and its relation to Kolmogorov backward/forward equations?

Kolmogorov backward/forward equations are pdes, derived for the semigroups constructed from the Markov transition kernels. Feynman-Kac formula is also a pde corresponding to a stochastic process ...
11
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1answer
350 views

Practical applications of the $L^p$ norm when $p \neq 1,2,\infty$

I'm roughly familiar with the concept of $L^p$ norms -- what they represent and how they are computed -- though I am far from educated in functional analysis in general. For reasons that are more or ...
11
votes
1answer
445 views

Every Hilbert space operator is a combination of projections

I am reading a paper on Hilbert space operators, in which the authors used a surprising result Every $X\in\mathcal{B}(\mathcal{H})$ is a finite linear combination of orthogonal projections. The ...
11
votes
2answers
2k views

How to prove that a bounded linear operator is compact?

I encountered a homework problem that says: If $A$ is a bounded linear operator from $X$ to $Y$. And $K$ is a compact operator from $X$ to $Y$, where $X$ and $Y$ are both Banach spaces, and Ran$(A)\...
11
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1answer
491 views

Stone-Weierstrass implies Fourier expansion

To prove the existence of Fourier expansion, I have to solve the following exercise, which supposedly follows from the Stone-Weierstrass theorem: Let $G$ be a compact abelian topological group ...
11
votes
3answers
243 views

Prove that there exist linear functionals $L_1, L_2$ on $X$

Let $X$ be a linear space, $p, q$ sublinear functionals on $X$, and $L$ a linear functional on $X$ such that $|L(x)| ≤ p(x) + q(x),$ for all $x ∈ X$. Prove that there exist linear functionals $L_1, ...
11
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1answer
440 views

Approximate spectral decomposition

A detailed attempt below. I am interested in effective and constructive computations for finding approximate spectral decompositions in some suitable format. Namely, let $A: H \rightarrow H$ be a ...
11
votes
2answers
181 views

average of maximal function is less than its infimum?

Let M be the dyadic Hardy-Littlewood maximal operator. Prove the following: there is a constant $C$ such that for any $f$, $$ \inf_{x\in I}Mf(x)\le C 2^k\inf_{x\in J} Mf(x) $$ where $I$ and $J$ are ...