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A sequence $(x_n)$ in a Hilbert space $H$ is weakly convergent to $l$ if for every element $v$ in $H$, the inner product of $x_n$ with $v$ converges to the inner product of $l$ with $v$ as $n$ approaches infinity. This is denoted by $x_n \rightharpoonup l$.

Conditions: $H$ is a Hilbert space.; $(x_n)$ is a sequence in $H$.; $l$ is an element of $H$.; $v$ is an arbitrary element of $H$.

A sequence $(x_n)$ in a Hilbert space $H$ is weakly convergent to $l$ if for every element $v$ in $H$, the inner product of $x_n$ with $v$ converges to the inner product of $l$ with $v$ as $n$ approaches infinity. This is denoted by $x_n \rightharpoonup l$.

Conditions: $H$ is a Hilbert space.; $(x_n)$ is a sequence in $H$.; $l$ is an element of $H$.; $v$ is an arbitrary element of $H$.

The Riesz representation theorem allows identifying elements of the dual space with elements of the Hilbert space $H$ itself via the inner product. This adapts the general definition of weak convergence (which uses dual space functions) to the inner product form used specifically in Hilbert spaces.

Conditions: $H$ is a Hilbert space.; The general definition of weak convergence involves continuous linear functionals from the dual space.

The Riesz representation theorem allows identifying elements of the dual space with elements of the Hilbert space $H$ itself via the inner product. This adapts the general definition of weak convergence (which uses dual space functions) to the inner product form used specifically in Hilbert spaces.

Conditions: $H$ is a Hilbert space.; The general definition of weak convergence involves continuous linear functionals from the dual space.

Strong convergence implies weak convergence because the Cauchy-Schwarz inequality bounds the difference of inner products by the product of the norm of the difference and the norm of the test vector. If the sequence converges strongly, the norm of the difference goes to zero, forcing the inner product difference to go to zero for any fixed test vector.

Conditions: $H$ is a Hilbert space.; $(x_n)$ is a sequence in $H$.; $(x_n)$ converges strongly to $l$ (i.e., $\|x_n - l\| \to 0$).; $v$ is an arbitrary element of $H$.

Strong convergence implies weak convergence because the Cauchy-Schwarz inequality bounds the difference of inner products by the product of the norm of the difference and the norm of the test vector. If the sequence converges strongly, the norm of the difference goes to zero, forcing the inner product difference to go to zero for any fixed test vector.

Conditions: $H$ is a Hilbert space.; $(x_n)$ is a sequence in $H$.; $(x_n)$ converges strongly to $l$ (i.e., $\|x_n - l\| \to 0$).; $v$ is an arbitrary element of $H$.

The Kakutani theorem states that every bounded sequence in a Hilbert space $H$ admits a subsequence that converges weakly in $H$. This provides a compactness principle for weak convergence in infinite-dimensional spaces.

Conditions: $H$ is a Hilbert space.; $(x_n)$ is a bounded sequence in $H$, meaning $\sup_n \|x_n\|_H < \infty$.

The Kakutani theorem states that every bounded sequence in a Hilbert space $H$ admits a subsequence that converges weakly in $H$. This provides a compactness principle for weak convergence in infinite-dimensional spaces.

Conditions: $H$ is a Hilbert space.; $(x_n)$ is a bounded sequence in $H$, meaning $\sup_n \|x_n\|_H < \infty$.

A sequence $(x_n)$ in a Hilbert space $H$ is strongly convergent to $l$ if the norm of the difference between $x_n$ and $l$ approaches zero as $n$ approaches infinity. This is denoted by $x_n \rightarrow l$.

Conditions: $H$ is a Hilbert space.; $(x_n)$ is a sequence in $H$.; $l$ is an element of $H$.

A sequence $(x_n)$ in a Hilbert space $H$ is strongly convergent to $l$ if the norm of the difference between $x_n$ and $l$ approaches zero as $n$ approaches infinity. This is denoted by $x_n \rightarrow l$.

Conditions: $H$ is a Hilbert space.; $(x_n)$ is a sequence in $H$.; $l$ is an element of $H$.

The Cauchy-Schwarz inequality is used to bound the absolute difference of the inner products $|\langle x_n, v \rangle - \langle l, v \rangle|$ by the product of the norm of the difference $\|x_n - l\|$ and the norm of the test vector $\|v\|$. This step converts the problem of showing inner product convergence into showing that a product involving the norm goes to zero.

Conditions: $H$ is a Hilbert space.; $(x_n)$ is a sequence in $H$ converging strongly to $l$.; $v$ is an arbitrary element of $H$.

The Cauchy-Schwarz inequality is used to bound the absolute difference of the inner products $|\langle x_n, v \rangle - \langle l, v \rangle|$ by the product of the norm of the difference $\|x_n - l\|$ and the norm of the test vector $\|v\|$. This step converts the problem of showing inner product convergence into showing that a product involving the norm goes to zero.

Conditions: $H$ is a Hilbert space.; $(x_n)$ is a sequence in $H$ converging strongly to $l$.; $v$ is an arbitrary element of $H$.