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Questions tagged [functional-analysis]

Functional analysis, 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 and other topics. For basic questions about functions use more suitable tags like (functions), (functional-equations) or (elementary-set-theory).

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, including spectral theory, as well as measure, integration, probability on infinite dimensions, and also manifolds with local structure modeled by these vector spaces.

For basic questions about functions use more suitable tags like , or .

52582 questions
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A question about conic sets of functionals.

The problem is the following. Let $(X,\|\cdot\|)$ be a normed space and let $C \subseteq X$ be a closed convex set with nonempty interior. Let be $x\in C$. I define to be a "normal cone to $C$ in $x$" the set: $N_C(x):=\{ f \in X^\ast : \langle…
Benzio
  • 2,097
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Semi-norms in Functional Analysis

I'm self-studying functional analysis. The following is from Rudin's "Functional Analysis, 2nd edition". It consists of parts from question 7 and 13 from the first chapter. I am not sure if my answers are correct, and would appreciate any…
MathsStudent
  • 179
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Non compact embedding

Could someone please explain to me, why the embedding $\iota \colon C^0( \overline{\Omega}) \to L^2(\Omega)$ is not compact? $\Omega=(0,1)$ and $ \overline{\Omega}$ denotes the closure of $\Omega$. I already got the hint to use the sequence…
David75
  • 383
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Solve the equation: $x(t)-3\int_0^1(s+t)x(s)ds=y(t)$

Given $y\in L^2[0,1]$, Solve the equation: $$x(t)-3\int_0^1(s+t)x(s)ds=y(t)$$ I have noticed that the equation is $(I-K)(x(t))=y(t)$, where $K(f(t))=\int_0^13(s+t)f(s)ds$ is a compact integral operator in $L^2[0,1]$, so Fredholm alternative is an…
user164777
  • 33
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Property of norm

Let $X$ be a compact Hausdorff space and let $C(X)$ denote the set of continuous complex valued functions on $X$. Define $$ \|f\|:=\sup\{|f(x)|:x\in X\},$$ then prove that $\|fg\|\leq \|f\|\|g\|$.
Nannes
  • 1,141
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Different definitions of Morrey and Campanato Spaces

The book by Giaquinta defines Campanato spaces using the seminorm: $$[u]_{p,\lambda} = \left(\sup_{\substack{{x_0\in\Omega \\ 0
Loreno Heer
  • 4,460
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Show that there exists constant $C$ such that $\sum_{n=1}^{\infty}|\langle f,x_n\rangle |^2\le C \|f\|^2$

Let $H$ be Hilbert space. I have to show that if $\sum_{n=1}^{\infty}|\langle f,x_n\rangle|^2 < \infty, \:\:\: f\in H$ then there exists constant $C\ge 0$ such that $\sum_{n=1}^{\infty}|\langle f,x_n\rangle|^2\le C \|f\|^2, \:\:\: f\in H$ Is this…
luka5z
  • 6,359
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Is the derivative of a bump function still a bump function?

My question is rather simple : are derivatives of bump functions still bump functions ? For example, for a bump function $u\in D(\mathbb{R}^d)$, that is, $$u \in C^\infty|\;\text{supp}\;u\subseteq K : \mathbb{R}^d \to \mathbb{R}$$ Is this always…
mwoua
  • 865
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Bound of direct sum of operators

Let $X$ be a Banach space and $U,V$ complementary subspaces. Let $A: U \to U, B: V \to V$ be continuous and let $T(x) = (A \oplus B)(u \oplus v)= A(u) \oplus B(v)$. Does it hold that $\|T\| \le \|A \| + \|B\|$?
Student
  • 1,687
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Adjoint of sum = sum of adjoints

is $\mathcal{D}(A)=\mathcal{D}(B)$ a sufficient condition for $(A+B)^*=A^*+B^*$ , where $A$ and $B$ are densely defined (not necessarily symmetric) operators on some Hilbert space?
user51527
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1 answer

Completion, injection

$\mathcal{H}$: Real hilbert space with inner product $(\cdot,\cdot)$. $D$ is a subspace of $\mathcal{H}$. Let $\mathcal{E}$ with domain $D$ be a positive definite (i.e. for all $u \in D $, $\mathcal{E}(u,u)\geq0$) bilinear form on $\mathcal{H}$. For…
ko4
  • 541
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$L^{\infty}(\mu)$ is $C(K)$ for some compact Hausdorff space $K$ when $\mu$ is sigma-finite

I am looking for a proof (or at least an article or book in which it is stated) that for measure space $(X, A, \mu)$, where $\mu$ is sigma-finite, there exists compact Hausdorff space K such that $L^{\infty}(\mu)$ is isometrically isomorphic to…
Adam
  • 141
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Can $C(\mathbb R)$ be reflexive?

As in title, I was wondering whether $C(\mathbb R)$ was reflexive (here $C(\mathbb R)$ is meant as the space of continuous functions on $\mathbb R$, without any other condition). This question is generated by the following well-known result:…
user91126
  • 2,326
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$L^2$-convergence of a sequence of step functions of differences

Suppose $f\in L^2=L^2([0,1])$ (that is, $f$ is square-integrable). Let $f_n$ be an approximation of $f$ by steps of differences of $\int f$. Formally, $$f_n =…
Bach
  • 2,177
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Weak Compactness & Separable Subspaces

The following is an easy Corollary of the equivalence of weak compactness and weak sequential compactness in the weak topology on a normed space $X$: A subset $E$ of a normed space $X$ is weakly compact if and only if $E\cap Y$ is weakly compact…
Student
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