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In the Hilbert space view, this is the orthogonal projection of onto the kernel of the expectation operator, which a continuous linear functional on the Hilbert space (in fact, the inner product with the constant random variable 1), and so this kernel is a closed subspace.
The first three functions in the sequence () = on [,].As converges weakly to =.. The Hilbert space [,] is the space of the square-integrable functions on the interval [,] equipped with the inner product defined by
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space [1] [2]) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar , often denoted with angle brackets such as in a , b {\displaystyle \langle a,b\rangle } .
In the complex case, the inner product on is defined by , = () ... As any Hilbert space, every space is linearly isometric to a suitable (), where the ...
Every inner product space is also a normed space. A normed space underlies an inner product space if and only if it satisfies the parallelogram law, or equivalently, if its unit ball is an ellipsoid. Angles between vectors are defined in inner product spaces. A Hilbert space is defined as a complete inner product space. (Some authors insist ...
The quotient space of by the vector subspace is an inner product space with the inner product defined by +, + := (),,, which is well-defined due to the Cauchy–Schwarz inequality. The Cauchy completion of A / I {\displaystyle A/I} in the norm induced by this inner product is a Hilbert space, which we denote by H {\displaystyle H} .
When the Hilbert space is reinterpreted as a real Hilbert space then it will be denoted by , where the (real) inner-product on is the real part of 's inner product; that is: , := , .
Again, H s,p (Ω) is a Banach space and in the case p = 2 a Hilbert space. Using extension theorems for Sobolev spaces, it can be shown that also W k,p (Ω) = H k,p (Ω) holds in the sense of equivalent norms, if Ω is domain with uniform C k -boundary, k a natural number and 1 < p < ∞ .