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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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How to make sense of this integral?

Let $L$ be the Ornstein-Uhlenbeck operator on $L^2(\gamma)$ where $\gamma$ is the Gaussian measure on $\mathbf R^d$. Hille-Yosida or Lumer-Phillips can be used to prove that $L$ generates a strongly continuous semigroup $e^{tA}$. Now I have the…
JT_NL
  • 14,514
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3 answers

Proving $\|T^n\|\leq \|T\|^n$

Let $T:X\to X$ be a linear bounded mapping. I have to prove $\|T^n\|\leq \|T\|^n$. Let $Tx=cx$, where $c>0$. This is a linear mapping. $$T^2 x=T(Tx)=T(cx)=cTx=c^2 x.$$ Hence $\|T^2x\|=c^2\|x\|.$ Similarly, $\|T^n x\|=c^n\|x\|$. $$\|Tx\|^n=c^n…
user67803
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Using inequalities

UPDATE: Let $x=(x_{n})$ and $y=(y_{n}) \in A$ with $A:=\{x=(x_{n})\in \ell^{2}| \phantom{x} \|x\|\leq 1\}$. Prove that $d:A\times A \rightarrow \mathbb{R}_{+}$ defined by $$d(x,y)=\sum_{n=1}^{\infty}(1/3)^{n}|x_{n}-y_{n}|$$ is bounded. Let…
Lech121
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$\mathcal{L}_{1}$ space criterion.

$\bf{\text{Definition:}}$ Let $X$ be a Banach space. $X$ is an $\mathcal{L}_{1,\lambda}$-space if, for all finite-dimensional subspaces $M$ of $X$, there exists a finite dimensional subspace $N$ of $X$ containing $M$, and an isomorphism $T\in…
roo
  • 5,598
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Problem with mapping concerning $c_{0}$

Show that the mapping $$ \Phi: \ell^{1}\rightarrow \mathcal{L}(c_{0};\mathbb{K}), \phantom{x} \Phi_{ x}(y):=\sum_{n=1}^{\infty}x_{n}y_{n}$$ is well-defined and an isometric isomorphism. Updated my answer: The functional $$ x \mapsto…
Lech121
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$\rho(T)=\rho(T^*) $ if $T$ is defined between Banach Spaces.

Let $T:E\rightarrow E$ be a continuous linear map between Banach spaces. We define $T^*:E^*\rightarrow E^*$ by $T^*(e^*)(e)=e^*(T(e))$. Under these conditions prove that the resolvent set $\rho(T)=\{\lambda \:| T-\lambda I \: \text{ is…
Kadmos
  • 1,907
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2 answers

"Adding one dimension" to an infinite dimensional topological vector space

Let $V$ be an infinite-dimensional Hausdorff topological vector space over $\mathbb{R}$. Is it true that $V\oplus\mathbb{R}$ is isomorphic (as a TVS) to $V$ itself? (How about $V\oplus V$?) If yes, I also wonder whether for every $v \ne 0$ in $V$…
Junyan Xu
  • 698
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Show this bounded linear operator is injective

Let $X=C([0,1])$ with the $\sup$ norm and let $T: X \to X$ given by $$ Tf(t) = f(t) + \int_0^t f(s) \,\mathrm{d}s. $$ I already showed that $T$ is a bounded linear operator with $\|T\|=2$. The problem asks me to show that $T$ is injective. What I…
EllaW
  • 307
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Showing that a set is complete in $L^2[0, \, \pi/2]$

I'm trying to show that the set $S = \{ \sin((2n - 1)x)\}$ for $n = 1, 2, \dots $ forms a complete system for the Hilbert space $L^2[0, \, \pi/2]$. In other words I have to show that if $f \in L^2[0, \, \pi/2]$ satisfies $$\langle f , \, \sin((2n -…
saurs
  • 1,377
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Kernel of the closure of an unbounded operator

Let $T:\mathcal{D}(T)\subset X\to Y$ be a densely defined closable operator. Define $\text{ker}\,T=\{(x,0)\in \text{graph}\,T\}$ where $\text{graph}\,T\subset X\times Y$. My question is $\overline{\text{ker}\,T}=\text{ker}\,\overline{T}$? I recall…
user90189
  • 1,630
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The Volterra Operator and the distance between a point and its range

Let $V:L^2[0,1]\to L^2[0,1]$ be the Volterra operator given by $f\mapsto V(f)$ where $$V(f)(t)=\int_0^tf(s)ds,\ \forall t\in[0,1].$$ My question is: Is it true that for for each $d>0$ small there exists $f\in L^2[0,1]$ such that $$\parallel…
Carlos
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Boundedness of Linear Functionals

I came across the Definition of Bounded linear functionals, that is: A linear functional $f$ defined on a normed space $X$ that is, $f: X\mapsto\mathbb{C}$ is said to be bounded if $\exists\, M>0$ such that $\forall x$ $\in X$, $$|f(x)|\le…
ANIMESH JAIN
  • 43
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Is projection on a closed subspace of a Banach space bounded?

We know that any projection on a closed subspace of a Hilbert space is bounded. Is it true for any Banach space? Any help would be appreciable.
Shailesh
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How to prove the continuity of the functional $v \mapsto \int \vert v \vert^2$ on this function space?

I am trying to prove the continuity of the functional \begin{equation} f: v \mapsto \int_{\mathbb{R}^N} \vert v \vert^2 \end{equation} on the space \begin{align} V(\mathbb{R}^N) = \lbrace v = v_1 + i v_2 : \mathbb{R}^N \to \mathbb{C} ~ | ~ &\nabla v…
mjb
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If $X$ is normed space, $V \subset X$ is closed, and $W \subset X$ is finite-dimensional with $V \cap W = \{0\}$, then $\pi(V) \subset X/W$ is closed

Let $X$ be a normed space, let $V \subset X$ be a closed subspace, and let $W \subset X$ be a finite-dimensional subspace with $V \cap W = \{0\}$. I would like to show that $\pi(V) \subset X/W$ is closed, where $\pi : X \to X/W$ is the quotient…
user1134586
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