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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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Bump function that is not piecewise defined?

Is there a practical example of a real, one-variable function with compact support that is not defined piecewise? Most computer algebra systems have a hard time dealing with such expressions, so a bump function that isn't piecewise defined would be…
Johnny
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What will happen when we move the time integration out of space norm?

Given $f(\mathbf{x},t)\in L^2\big((t_1,t_2);\mathbf{L}^2(\Omega)\big)$, how to prove the following inequality? $$ \Bigg\|\int_{t_1}^{t_2}f(\mathbf{x},t)dt\Bigg\|_{\mathbf{L}^2(\Omega)} \le…
chentaocuc
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Extending a non-compact operator to a non-compact operator between Hilbert spaces

Let $X,Y$ be separable Banach spaces, and $S: X \rightarrow Y$ be a non-compact (continuous) operator. Can I always find two separable Hilbert spaces $H_1, H_2$ and continuous operators $T: H_1 \rightarrow X, T_2: Y \rightarrow H_2$ such that $$T_2…
Marcin Wnuk
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Question in infinite- dimensional banach space.

let $E$ be a banach space with infinite dimensional and $F$ a subspace of $E$. is it true that if $F^° \neq \emptyset $ then $E=F$. my attempt: $F$ a subspace of $E$ then $F\subset E$ we have to prove that $E\subset F$ i think we should use the…
Geravit3
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Application of Hahn-Banach

Let $(E,\mathcal{E},\mu)$ be a measured space with finite measure $\mu$. We denote with $K$ the space of all real valued functions on $E$, which are $\mu$-a.s. equal. This is a vector space. Now I have a function $\kappa:K\to \mathbb{R}$, which…
user20869
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Prove: Edelstein's fixed point theorem

Let X be a metric space. A self-map $\Phi$ on X is said to ve a pseudocontraction if $d(\Phi(x),\Phi(y))
peter xue
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Kernel of an unbounded linear functional on Banach space

Assume $f$ is an unbounded linear functional on Banach space $X$. Then $\ker(f)$ is a dense linear subspace of $X.$ Is $\ker(f)$ a set of second category ?
Ryszard Szwarc
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Properties of a functional on $\ell^\infty$

For $\Phi: \ell^\infty \to \mathbb R$ a linear map, consider the following three properties: (a): $\Phi$ sends the constant sequence $(c, c, \ldots)$ to $c$; (b): $\Phi(x)\ge 0$ whenever $x \ge 0$ (c): $| \Phi(x)|\le \| x \|.$ How to prove that…
arnett
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Theorem 1.14 in Rudin's book

Theorem 1.14. In a topological vector space $X$, (a) every neighborhood of $0$ contains a balanced neighborhood of $0$, and (b) every convex neighborhood of $0$ contains a balanced convex neighborhood of $0$. Proof: (a) Suppose $U$ is a neighborhood…
Mingg
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Separation by functional in a subspace of the dual space

Let $X$ be a locally convex topological vector space with topology $\tau$. Let $X^*$ be the set of linear functionals on $X$ endowed with the weak* topology. Let $Z^* \subset X^*$ be a vector subspace such that the topology $Z^*$ induces on $X$ is…
Smithey
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$\dim\ker T=\dim\ker T^*+1$ for $(Tx)(m)=x(m+1)+\sum_{n\geq1}a_{m,n}x(n)$ on $\ell^2$

Let $(a_{m,n})$ be a double sequence of complex numbers such that $\sum_{m\geq1}\sum_{n\geq1}|a_{m,n}|^2<\infty$, and define the bounded linear map $T:\ell^2\to\ell^2$ by $$(Tx)(m)=x(m+1)+\sum_{n\geq1}a_{m,n}x(n).$$ I'm asked to prove that $\dim\ker…
Václav Mordvinov
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Functional analysis and Banach Algebras well defined

I have a question regarding well - definedness. Suppose $X$ is a banach space $\mathcal{l}^{1}(\mathbb{Z})$ given by the norm $||(x_{n})_{n}||_{1} := \sum_{n \in \mathbb{Z}} |x_{n}|$ If we define the product $xy$ as $(xy)_{n} = \sum_{m}…
Chengdu
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confusion about holomorphic functional calculus

Let $H$ be a Hilbert space and $T:H\rightarrow H$ is a bounded linear operator. By holomorphic functional calculus, we can write $\displaystyle T=\oint_{C}\frac{z}{z-T}dz$, where $C$ is the contour enclosed the spectrum of $T$. Now choose…
Ken.Wong
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clarification for invariant subspace problem

According to what I know, invariant subspace problem asks for if any operator of Banach space has a nontrivial closed invariant subspace. I am not sure what "nontrivial" means here. Of course $\{0\}$ and the whole space count as trivial subspace.…
Ken.Wong
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1 answer

Are functions $f \in C^\infty[0,1], f^{(n)}(0)=f^{(n)}(1)=0 \ \forall n\geq0$ dense in $L^2[0,1]$?

Suppose that $X \subset L^2[0,1]$ is the set $$ X = \{f \in C^\infty[0,1] : f^{(n)}(0)=f^{(n)}(1)=0 \ \forall n\geq0 \}. $$ Is this a dense subspace of $L^2[0,1]$? I know that the space of continuous functions is dense in $L^2[0,1]$. My approach…
Muzi
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