# 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).

32,000 questions
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### To Prove $T$ is a self map and $T$ have no fixed points

Let $K=\{x=(x(n))\in c_0:0\le x(n)\le 1$ for all $n\in \mathbb{N}\}$. Define $T:K\to c_0$ by $T(x)=(1,x(1),x(2),x(3),...).$ Prove : (a) $T$ is a self map on $K$ and $||Tx-Ty||_\infty=||x-y||_\infty$ ...
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### Is this a Projection operator on Hilbert space?

Let $T$ be a bounded operator on a Hilbert space with the property that $T^*(T-I)= 0$. I'd like to show that $T$ is an orthogonal projection. I'm not really sure how to show that an operator is an ...
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### Show that $f_n$ converges in $L^1$ norm

Suppose that $f_n: X \to C$ are a dominated sequence of measurable functions, and let $f:X \to C$ be another measurable function. Show that $f_n$ converges in $L^1$ norm to $f$ if and only if $f_n$ ...
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### Example of Drazin invertible operator that is not invertible

Let $X$ be a Banach space. A bounded operator $T$ in $X$ is said to be Drazin invertible if there exists $k \in \mathbb{N}$ and a bounded operator $T^D$ in $X$ such that $TT^D = T^DT$...
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### On the Schrodinger fundamental solution

Let $e^{it\Delta}$ be the fundamental Schrodinger solution. If $u_0$ is the corresponding initial data to the problem associated to Schrodinger free equation $u_t = i\Delta u$ and $S(\mathbb{R}^N)$ ...
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### Special elements in the $C^*$ algebra $A \otimes \mathcal{K}$.

Context: Let $A$ be an ungraded (not necessarily unital) $C^*$ algebra. $\mathcal{K}$ space of compact bounded operators on an infinite separable graded Hilbert space $H=H_0 \oplus H_1$. Consider ...
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### n-times commutator of a function with $-\Delta +v$

Let $f$, $v$ be smooth function and let us assume that $v$ is also bounded. I'm needing somehow a formula for the $n$-times commutator of $f$ (interpreted as multiplication operator) with the ...
### If $U\subset W\subset V$ and $W\cap U^\perp = \{0\}$ then $U=W$
Statement : Let $V$ be a Hilbert space. Let $U\subset W\subset V$ be closed subspaces. Suppose that $W\cap U^\perp = \{0\}$. Then $U=W$. I know this is true in the finite dimensional case (see proof ...