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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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Question about quotient of separable spaces

Exercise: Let $E$ a Banach space separable and $F$ a closed subspace of $E$. Prove that $E/F$ is separable. My idea: Since $E$ is separable then exist $D_1\subset E$ numerable and dense and Since F is closed then $F=\bar{F}$. Moreover $F$ is…
MCM
  • 85
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Series Expansion of Exponential Function is Cauchy

Define $p_n (x) = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \dots + \frac{x^n}{n!}$. Show that $\{ p_n (x) \}_{n=0}^{\infty}$ is a Cauchy sequence. I'd just like to know the facts required to show this. We know that if a function $f: A \mapsto…
Jared
  • 845
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Use functional analysis to show that integral equation has unique solution

Let $b>a$ in $\mathbb R$ and $K:[a,b]\times[a,b] \rightarrow \mathbb C$ be a continuous function. Show that for all $\lambda \ne 0$ in $\mathbb{C}$ and $f \in C[a,b]$ the integral equation $$ \int_a^t K(t,s) u(s) ds - \lambda u = f$$ has one…
José
  • 953
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Meaning of the notation $\int_{\mathbb R}^\oplus P(x) dx$ and $L^2(\mathbb R, H)$

While reading the book "Adiabatic Perturbation Theory in Quantum Dynamics" by Teufel, I came up with the following notations: $$P=\int_{\mathbb R^{3l}}^\oplus dx P_(x)$$ I searched Wikipedia and found out that this notation is about the "direct…
Laplacian
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If the spectral radius of an operator $T$ is $>1$, show that $\sum_{j=0}^\infty$ diverges.

Let $X$ be a Banach space, $T: X \rightarrow X$ a bounded operator and $r_\sigma (T) = \lim_{n \rightarrow \infty} \|T^n\|^{1/n}$ the spectral radius of $T$. Show that if $r_\sigma (T) > 1$ then the series $\sum_{j=0}^\infty T^j$ does not…
José
  • 953
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Inner product space not the direct sum of subspace and orthogonal complement?

Here is a problem I'm working on: If $M$ is a closed subspace of an inner product space $V$, is it always true that $V=M\oplus M^\perp$? I am asked to consider the following example: Let $V\subset\ell^2$ be the set of all sequences with only…
buffle
  • 623
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Show that every compact operator has non-separable kernel

Let $H$ be a non-seperable Hilbert space. Show that every compact operator $T: H \rightarrow H$ has non-separable kernel. Since $T$ is compact them the image of the unity ball should have compact closure. This doesn't tell me anything about the…
José
  • 953
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$\{f \in C^1([0,1]): \|f\|_\infty + \|f'\|_\infty \leq \gamma\}$ is a compact subset of $C[0,1]$

A textbook I'm reading states the set $A = \{f \in C^1[0,1]: \|f\|_\infty + \|f'\|_\infty \leq \gamma\}$ is compact in $C[0,1]$ with the sup norm. Here, $\gamma$ is a positive constant. Well, by Arzela Ascoli, as this set is equicontinuous and…
Anu
  • 1,257
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$\varphi(x)=0$ for every continuous linear form implies that $x=0$

Let $(E,||~~||)$ a normed vector space and $x\in E$. I want to show show that $$\varphi(x)=0~~\forall\varphi\in E'\Rightarrow~~x=0$$ Thank you in advance.
Donnie Darko
  • 501
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Example of Schauder basis to $\ell^2$

Let $\alpha, \beta \in \mathbb C - \{0\}$ such that $0 < |\frac{\beta}{\alpha}| < 1$ and $$ x_1 = (\alpha, \beta, 0,...), x_2 = (0, \alpha, \beta, 0,...), x_3 = (0, 0, \alpha, \beta, 0,...),... $$ Show that $(x_n)_{n \ge 1}$ is a Schauder basis of…
JuanDa
  • 159
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Proving if A and B are bounded operators such that A, B, and AB are normal, then BA is also normal.

From this problem we know that if $A$ and $B$ are matrices such that $A, B$, and $AB$ are normal, then $BA$ is also normal. I would like to know if this property holds in a more general way: if $A$ and $B$ are bounded operators in a Hilbert space,…
J.Doe
  • 41
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Is the limit of non-invertible bounded linear operators non-invertible?

Let $L(t)$, $t\in\mathbb{R}$ be a family of bounded linear operators, where $L(t):X\to X$ for some Banach space $X$. Let $t_n\to t$, and $L(t_n)$ converge to some $L(t)$ in the operator norm. If $L(t_n)$ is non-invertible for all $n$, is it true…
Phil
  • 1,078
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Is every weakly complete normed space (strongly) complete?

A normed space $X$ is said to be weakly complete if every weakly Cauchy sequence weakly converges in $X$. Is there any example of a space which is weakly complete, but not complete?
F.Flower
  • 23
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norm of product of normed spaces

If $(X_1,||.||_1)$ and $(X_2,||.||_2)$ are two normed spaces and define norm on $X_1\times X_2$ as $||x||=\max(||x_1||_1,||x_2||_2)$. I want to check the triangle inequality property for this norm, i.e, $||x+y||\leq||x||+||y||$. After using using…
Vaolter
  • 1,711
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Continuous inverse of an injective linear function

$X,Y,Z$ are Banach spaces, $A:X\to Y$ and $B:X\to Z$ are continous linear injective functions and $B$ is also compact. Moreover, there exists $C>0$ such that $\Vert x\Vert\leqslant C\Vert Ax\Vert+C\Vert Bx\Vert$. To show is there exists some…
CO2
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