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In this paper, we show that every classical inner product on a linear space induces the fuzzy inner product and fuzzy norm in the sense of Bag and Samanta. We prove the Cauchy-Schwarz inequality on fuzzy Hilbert spaces. We also define fuzzy frame on fuzzy Hilbert spaces. As is known, Cю (Ω) is not normable in classical Hilbert space, but in this paper we show that Cю (Ω) is normable in fuzzy Hilbert space and so defining fuzzy frame on Cю (Ω) is possible.