# 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).

52,923 questions
Filter by
Sorted by
Tagged with
27 views

### Notation for a sum of Hilbert spaces

Let $\mathcal{T}$ and $\mathcal{P}$ be a finite- and an infinite dimensional Hilbert spaces. Does the notation $\mathcal{T} + \mathcal{P}$ stand for \mathcal{S}=\mathcal{T} + \...
• 317
1 vote
39 views

### Numerical radius norm is equal to the operator norm

Currently, I am studying numerical ranges and numerical radius of linear operators. As one of the references, I am studying the book Numerical Range: The Field of Values of Linear Operators and ...
• 151
26 views

39 views

### Bounded operators on the span of a set of vectors

Let $E$ be a Banach space and $S$ a subset of vectors in $E$. Suppose that a linear operator $T$ satisfies the property $$\|Tx\| \leq C \|x\| \text{ for all }x \in S.$$ Does this imply that $T$ is ...
1 vote
18 views

### Dimension of a linear subspace invariant under closure.

Let $X$ be an infinite dimensional normed vector space over $\mathbb R$ and $W \subseteq X$ an $m$-dimensional subspace. Is it true that the topological closure $\overline{W} \subseteq$ is also an $m$-...
• 119
1 vote
38 views

• 1,181
17 views

### Berge's Maximum Theorem for Parameters that are Functions

I am given some functions $f_i$ that are $C^1$, $f_i : \mathbb{R}_+ \rightarrow \mathbb{R}_+$ and some budget $z \in \mathbb{R}_+$. I have some constrained optimization problem with a continuous ...
• 23
46 views

### Support of the Fractional Stationary Ornstein Uhlenbeck process (first kind)

I am interested in the support of the fractional stationary Ornstein Uhlenbeck process (first kind). In particular, I want to know whether smooth functions (or even only smooth functions starting at 0)...
• 21
48 views

• 131
1 vote
20 views

### Measurable in one variable and regular in the other

Suppose $(X,\mathcal{B},\mu)$ is a measure space and for $y\in\mathbb{R}$ suppose that the function $f(\cdot,y):X\to X$ is measurable and $f(x,\cdot)$ is continuous. In many cases, I have seen that ...
• 1,126
27 views

• 639
1 vote
25 views

### bv space is a direct sum of $bv_0$ with a one dimensional subspace

We define $bv=\{x=\{x_n\} | \sum_{k=1}^\infty |x_{k+1} -x_k| < \infty , x_k \in \mathbb C\}$ $C_0=\{x=\{x_n\} | lim x_n = 0,x_k \in \mathbb C\}$ And $bv_0=bv \cap c_0$ I need to prove that bv ...
• 131
16 views

### Understanding the transpose of a (generalized) Calderon-Zygmund operator

I have been reading Fourier Analysis by Javier Duoandikoetxea, and I have a question about generalized Calderon-Zygmund operators. For reference, I state the definition as given in the book. ...
• 4,047
91 views

• 1,079
26 views

### The $L^\infty$ boundedness of the resolvent of Harmonic Oscillator in terms of seminorms

I am reading Watanabe's book "Lectures on Stochastic Differential Equations and Malliavin Calculus". In Page 48, it is said that by means of (damped) harmonic oscillator $1+|x|^2-\Delta$, we ...
• 313
1 vote